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\begin{document}
\begin{frontmatter}
\title{Five-wave interaction on the surface of deep fluid\\ {\large Published in Physica D {\bf 87}, 233 (1995)}}

\author{A.I.Dyachenko$^{1,2}$, Y.V.Lvov$^{2,3}$ and
V.E.Zakharov$^{1,2}$}
\address{
$^1$ Landau Institute for Theoretical Physics, Russia, 117334, Moscow, Kosygina st.,2\\
$^2$ Department of Mathematics, University of  Arizona, Tucson, AZ, 85721,USA\\
$^3$ Department of Physics, University of  Arizona, Tucson, AZ, 85721,USA}
\begin{abstract}
The article deals with the studying of the interaction of gravity
waves propagating on the surface of an ideal fluid of infinite depth.
The system of the corresponding equation is proven to be integrable
up to the fourth order in power of stepness of the waves, but to be
nonintegrable in the next, fifth, order. The exact formula for the
five-wave scattering matrix element is obtained using diagram technique
on the resonant surface. The stationary solutions of the
five-wave kinetic equation are studied as well.
\end{abstract}
\end{frontmatter}

\section{Introduction}
\setcounter{equation}{0}
In this article we study interaction of gravity waves propagating in
one direction on the surface of ideal fluid of infinite depth. The
problem is of a big theoretical and practical importance. It is known
from the experiment that the distribution function of wave energy
even in the active zone of a storm is almost one-dimensional in the
energy-containing domain. Even more this is correct for the ``swell''
far away from the active zone. That is the point of common belief that the
main mechanism of wave interaction is the four-wave scattering,
satisfying the following resonant conditions
\begin{eqnarray} \label{res4}
\omega_{k} + \omega_{k_1} &=& \omega_{k_2} + \omega_{k_3} \cr
\vec k + \vec k_1 &=& \vec k_2 + \vec k_3
\end{eqnarray}
\noindent here $\vec k_i$ are the wave vectors of the interacting waves,
and $\omega_k = \sqrt{gk}$ - the dispersion law. The corresponding
effective Hamiltonian has the form
\begin{eqnarray} \label{ham4}
\Ha &=& \int \omega_k a_k a_k^* dk+\cr
&+& \frac{1}{4}\int T^{k k_1}_{k_2 k_3}
a_{k}^*a_{k_1}^*a_{k_2}a_{k_3}
\delta_{k+k_1-k_2-k_3}dkdk_1dk_2dk_3 + \ldots
\end{eqnarray}
\noindent ($a_k$ are complex amplitudes of propagating waves
\cite{Z68},\cite{CYS80},\cite{Z92}) and the corresponding kinetic
equation is
\begin{eqnarray} \label{kin4}
\frac{\partial n}{\partial t} &=& \pi \int |T^{k k_1}_{k_2 k_3}|^2
\delta_{k+k_1-k_2-k_3}
\delta_{\omega_{k}+\omega_{k_1}-\omega_{k_2}-\omega_{k_3}}\times \cr
&\times& \{n_{k_2}n_{k_3}(n_k + n_{k_1}) - n_k n_{k_1}(n_{k_2}n_{k_3})\}
dk_1dk_2dk_3
\end{eqnarray}
\noindent This equation is exactly equivalent (see \cite{DL94})
to the Hasselmann's
equation, derived first in 1962 \cite{H62}. The equation (\ref{kin4})
is entirely adequate to the situation if $\vec k_i$ are two-dimensional
vectors. But it completely fails in the one-dimensional case.

The equations (\ref{res4}) have in one-dimensional case two types of
solutions:
\par 1. The trivial solutions:
\begin{eqnarray} \label{triv4}
k_2 &=& k_1,\hspace{.5cm} k_3 = k,\cr
or\hspace{.5cm}k_3 &=& k_1,\hspace{.5cm} k_2=k.
\end{eqnarray}
\noindent Here $k$ and $k_1$ can have same or opposite signs.
\par 2. The nontrivial solutions ($k_2 \neq k, k_1$).
\par \noindent  These solutions exist only if the products $k k_1$ and
$k_2 k_3$ have opposite signs. They can be described analytically as
follow. Let $k > 0$, $k_1 > 0$, $k_2 < 0$, $k_3 > 0$. Then
\begin{eqnarray} \label{nontriv4}
k   &=& a(1+\zeta)^2, \hspace{0.5cm}k_1 = a(1+\zeta)^2 \zeta^2, \cr
k_2 &=& -a\zeta^2, \hspace{1.0cm}k_3 = a(1+\zeta+\zeta^2)^2
\end{eqnarray}
If one of the conditions (\ref{triv4}) holds, the expression
\begin{eqnarray} \nonumber
n_{k_2}n_{k_3}(n_k + n_{k_1}) - n_k n_{k_1}(n_{k_2}n_{k_3})
\end{eqnarray}
\noindent is equal to zero. So, the trivial solutions do not put
any contribution to the kinetic equation (\ref{kin4}). It is irrelevant
if all the wave numbers have the same sign (waves propagate in the same
direction). But even for waves propagating in the opposite directions,
four-wave interaction vanishes. As it was shown by Dyachenko and Zakharov
\cite{DZ94}, the coefficient $T^{k k_1}_{k_2 k_3}$ is identically equal to
zero
\begin{eqnarray} \nonumber
T^{k k_1}_{k_2 k_3} \equiv 0
\end{eqnarray}
\noindent on the manifold (\ref{nontriv4}).

This remarkable identity means that the system (\ref{ham4}) is
approximately integrable and kinetic equation appears for the next
order only
\begin{eqnarray} \nonumber
\frac{\partial n}{\partial t} = st(n,n,n,n).
\end{eqnarray}
\noindent Here $st(n,n,n,n)$ is the collision term due to five-wave
interaction, which are governed by the following resonant conditions
\begin{eqnarray} \label{res5}
\omega_{k} + \omega_{k_1} + \omega_{k_2} &=& \omega_{k_3} + \omega_{k_4} \cr
k + k_1 + k_2 &=& k_3 + k_4
\end{eqnarray}
\noindent A corresponding Hamiltonian has a form
\begin{eqnarray} \nonumber
&\Ha& = \int \omega_k a_k a_k^* dk +\cr
&+& \frac{1}{12}\int T^{k k_1 k_2}_{k_3 k_4}
\{a_{k}^*a_{k_1}^*a_{k_2}^*a_{k_3}a_{k_4}+c.c.\}
\delta_{k+k_1+k_2-k_3-k_4}dkdk_1dk_2dk_3dk_4 + \ldots
\end{eqnarray}
\noindent The expression $st(n,n,n,n)$ looks like that
\begin{eqnarray} \label{stoss5}
&&st(n,n,n,n) = \frac{\pi}{3}\int
|T^{k_1 k_2 k_3}_{k k_5}|^2
f_{k_1 k_2 k_3 k k_5}dk_1dk_2dk_3dk_5 - \cr
&-&\frac{\pi}{2}\int
|T^{k k_2 k_3}_{k_4 k_5}|^2
f_{k k_2 k_3 k_4 k_5}dk_2dk_3dk_4dk_5,\cr
&&f_{k_1 k_2 k_3 k_4 k_5} = \delta_{k_1+k_2+k_3-k_4-k_5}
\delta_{\omega_{k_1}+\omega_{k_2}+\omega_{k_3}
-\omega_{k_4}-\omega_{k_5}}\times \cr
&\times& \{ n_{k_1}n_{k_2}n_{k_3}(n_{k_4}+n_{k_5}) -
n_{k_4}n_{k_5}(n_{k_1}n_{k_2}+n_{k_1}n_{k_3}+n_{k_2}n_{k_3})\}
\end{eqnarray}
\noindent The expression (\ref{stoss5}) was found by Krasitskii \cite{KRS94}.
He also found ``in principle''  the expression for $T^{k k_1 k_2}_{k_3 k_4}$.
But his final formula is extraordinary complicated and cumbersome and hardly
can be used for any practical purpose. He used a technique of the  canonical
transformation which exclude gradually the low order nonlinear terms in the
Hamiltonian.

In this article we evaluate the coefficient $T^{k k_1 k_2}_{k_3 k_4}$ on the
resonant surface (\ref{res5}). Our final formulae are astonishingly simple.
This is one more miracle  in the theory of surface waves. We also find the
Kolmogorov's solution of the stationary equation
\begin{eqnarray} \nonumber
st(n,n,n,n) = 0
\end{eqnarray}
\noindent We will use a different technique then was used in
\cite{Z68},\cite{CYS80},\cite{Z92},\cite{H62}.


\section{Conformal canonical variables}
\setcounter{equation}{0}
The basic set of equations describing a two-dimensional
potential flow of an ideal incompressible
fluid with a free surface in a gravity field fluid is
the following one:
\begin{eqnarray} \nonumber
\phi_{xx} + \phi_{zz} &=& 0 \hspace{1cm} (\phi_z\to 0, z\to -\infty), \cr
\eta_t + \eta_x\phi_x &=& \phi_z\bigg|_{z=\eta}\cr
\phi_t + \frac{1}{2}(\phi_x^2 + \phi_z^2) + g\eta &=& {0\bigg|_{z=\eta}};
\end{eqnarray}
here $\eta(x,t)$ - is the shape of a surface, $\phi(x,z,t)$ - is a potential
function of the flow and $g$ - is a gravitational constant.
	As was shown by Zakharov in\cite{Z68},
potential on the surface $\psi(x,t) = \phi(x,z,t)\bigg|_{z=\eta}$
and $\eta(x,t)$ are canonically conjugated,
and their Fourier transforms satisfy the equations

\[\frac{\partial \psi_k}{\partial t} =
-\frac{\delta \Ha}{\delta \eta_k^*} \hspace{2cm}
  \frac{\partial \eta_k}{\partial t} = \frac{\delta \Ha}{\delta \psi_k^*}.\]

\noindent Here $\Ha=K+U$ is the total energy of the fluid with the following
kinetic and potential energy terms:

\begin{eqnarray} \nonumber
   K = \frac{1}{2}\int\!dx\!\int_{-\infty}^\eta\,v^2\!dz \hspace{1cm}
   U = \frac{g}{2}\int \eta^2\!dx
\end{eqnarray}

\noindent A Hamiltonian can be expanded in an infinite series in powers of
a characteristic wave steepness $k\eta_k <\!< 1$ (see\cite{{Z68},{CYS80}})
by using iterative procedure. All terms up to the fifth order of this series
contribute to the amplitude of five-wave interaction.

Here we prefer to do that performing first a certain canonical transformation
from the variables $\psi$, $\eta$ to the new canonical variables.

Let us perform, following Kuznetsov, Spector  and Zakharov \cite{KSZ94}, a
conformal mapping of the domain $z < \eta(x,t)$ to the lower half-plane of
the complex variable $w = u + iv, -\infty<u<\infty, -\infty<v<0$. The shape
of the surface is parametrized by two functions
\begin{eqnarray}  \nonumber
z(u,t), \hspace{0.5cm} x(u,t)
\end{eqnarray}
\noindent which are connected by the Hilbert transformation
\begin{eqnarray} \nonumber
   x(u,t) &=& u - \hat H(z(u,t))\cr
   \hat H(f(u)) &=& \frac{1}{\pi}P.V.\int \frac{f(u^\prime)du^\prime}
   {u^\prime-u}
\end{eqnarray}
\noindent We introduce also the complex velocity potential
\begin{eqnarray} \nonumber
\Phi(w,t) = \Psi(u,v,t)+i\Theta(u,v,t)
\end{eqnarray}
\noindent On the surface ($v=0$)
\begin{eqnarray} \nonumber
   \Theta(u,0,t) = \hat H(\Psi(u,0,t)), \cr
\end{eqnarray}
\noindent New canonical variables can be obtained using variational
principle for the action. With old variables action is
\begin{eqnarray} \nonumber
   S = \int dt \{ \int \psi(x,t)\eta_t(x,t)dx - \Ha\}.
\end{eqnarray}
\noindent After conformal mapping\cite{KSZ94} it acquires the form:
\begin{eqnarray} \nonumber
   S &=& \int L dt, \cr
   L &=& \int \{\Psi(z_t x_u - x_t z_u) + \frac{1}{2}\Psi \hat H\Psi_u
                -\frac{g}{2} z^2 x_u \}du.
\end{eqnarray}

\noindent Lagrangian function $L$ can be rewritten as
\begin{eqnarray} \nonumber
   L = \int \{z_t(\Psi x_u - \hat H(\Psi z_u)) +
  \frac{1}{2}\Psi \hat H\Psi_u - \frac{g}{2} y^2 x_u \}du.
\end{eqnarray}
and the new canonical variables are $z(u,t)$ and $\Pa(u,t) =
\Psi x_u - \hat H(\Psi z_u)$. $\Psi$ can be easily inverted as
\begin{eqnarray} \label{PsiPa}
   \Psi = \frac{\Pa x_u + \hat H(\Pa z_u)}{x_u^2+z_u^2}
\end{eqnarray}

\noindent and the Hamiltonian of the system is
\begin{eqnarray} \nonumber
   \Ha = \frac{1}{2} \int \{g z^2 x_u - \Psi \hat H\Psi_u\} du
\end{eqnarray}
\noindent where $\Psi$ is equal to (\ref{PsiPa}). The equations
of motion can be written in the explicit Hamiltonian form which
includes integral Hilbert's operator:

\begin{eqnarray} \nonumber
\frac{\partial \Pa}{\partial t} =
-\frac{\delta \Ha}{\delta z} \hspace{2cm}
  \frac{\partial z}{\partial t} = \frac{\delta \Ha}{\delta \Pa}
\end{eqnarray}

\section{Perturbation expansion for the Hamiltonian}
\setcounter{equation}{0}

One can introduce new variable $\tilde x$ as
\begin{eqnarray} \nonumber
x=u+\tilde x,\hspace{0.3cm} x_u  = 1+\tilde x_u, \hspace{0.3cm}
\tilde x = -\hat Hz
\end{eqnarray}
\noindent Then
\begin{eqnarray} \nonumber
   \Psi = \frac{\Pa + \Pa \tilde x_u + \hat H(\Pa z_u)}{(1+\tilde x_u)^2+z_u^2}
\end{eqnarray}

Now one can expand $\Psi$ in powers of $\tilde x_u$ and $z_u$
\begin{eqnarray} \nonumber
\Psi &=& \Psi^{(1)} + \Psi^{(2)} + \Psi^{(3)} + \Psi^{(4)} + \ldots \cr
\Psi^{(1)} &=& \Pa \cr
\Psi^{(2)} &=& \hat H(z_u \Pa) - \tilde x_u \Pa \cr
\Psi^{(3)} &=& \Pa({\tilde x_u}^2-z_u^2)-2\tilde x_u \hat H(z_u \Pa)\cr
\Psi^{(4)} &=& \Pa \tilde x_u (3z_u^2-{\tilde x_u}^2) +
                 (3{\tilde x_u}^2 - z_u^2)\hat H(z_u \Pa)
\end{eqnarray}
\noindent The Hamiltonian of the system is
\begin{eqnarray} \nonumber
   \Ha = \frac{1}{2} \int \{g z^2 (1+\tilde x_u) - \Psi \hat H\Psi_u\} du
\end{eqnarray}
\noindent Now $\Ha$ can be expanded as follow
\begin{eqnarray} \label{HamSer}
\Ha = \Ha_2 + \Ha_3 + \Ha_4 + \Ha_5 + \ldots
\end{eqnarray}
\noindent here
\begin{eqnarray} \nonumber
\Ha_2 &=& {1\over2}\int(g z^2 - \Psi^{(1)} \hat H\Psi^{(1)}_u) du \cr
\Ha_3 &=& {1\over2}\int(g z^2\tilde x_u -2\Psi^{(2)}\hat H\Psi^{(1)}_u) du \cr
\Ha_4 &=& -{1\over2}\int (2\Psi^{(3)} \hat H\Psi^{(1)}_u +
\Psi^{(2)} \hat H\Psi^{(2)}_u ) du \cr
\Ha_5 &=& -\int (\Psi^{(4)} \hat H\Psi^{(1)}_u +
\Psi^{(3)} \hat H\Psi^{(2)}_u ) du
\end{eqnarray}
\noindent Let's introduce the Fourier transform:
\[f_k = {1\over \sqrt{2\pi}} \int f(u) e^{-i k u} du,\hspace{1cm}
f(u) = {1\over \sqrt{2\pi}} \int f_k e^{i k u} dk\]
\noindent After simple, but a little bit tedious calculations one
can find
\begin{eqnarray} \nonumber
\Ha_2 &=& \frac{1}{2}\int (g |z_k|^2 + |k||\Pa_k|^2) dk, \cr
%
\Ha_3 &=&\frac{1}{2\sqrt{2\pi}} \int z_{k_1}\{ S_{k_1 k_2 k_3}
        z_{k_2}z_{k_3} - F^{k_1}_{k_2 k_3}
\Pa_{k_2} \Pa_{k_3} \} \delta_{k_1+k_2+k_3}dk_1dk_2dk_3, \cr
\Ha_4 &=& \frac{1}{4\pi}\int\!
M^{k_1 k_2}_{k_3 k_4}z_{k_1}z_{k_2}\Pa_{k_3}\Pa_{k_4}
\delta_{k_1+k_2+k_3+k_4}dk_1dk_2dk_3dk_4, \cr
\Ha_5 &=& \frac{1}{2(2\pi)^{3\over 2}}\int\!N^{k_1 k_2 k_3}_{k_4 k_5}
z_{k_1}z_{k_2}z_{k_3}\Pa_{k_4}\Pa_{k_5}
\delta_{k_1+k_2+k_3+k_4+k_5}dk_1dk_2dk_3dk_4dk_5
\end{eqnarray}
\noindent Here $S_{k_1 k_2 k_3}$, $F^{k_1}_{k_2 k_3}$, $M^{k_1 k_2}_{k_3 k_4}$
and $N^{k_1 k_2 k_3}_{k_4 k_5}$ are the functions symmetric inside of
upper and lower groups of indices. Namely
\begin{eqnarray} \label{SLF}
S_{k_1 k_2 k_3} &=& {g\over 3}(|k_1|+|k_2|+|k_3|)\cr
L_{k_1 k_2} &=& k_1 k_2 +|k_1 k_2|\cr
F^{k_1}_{k_2 k_3} &=& L_{-k_1 k_2} + L_{-k_1 k_3} = |k_1|(|k_1|+|k_2|+|k_3|)
\end{eqnarray}
\noindent The expression for $M^{k_1 k_2}_{k_3 k_4}$ and
$N^{k_1 k_2 k_3}_{k_4 k_5}$ are given in the Appendix A.

It is convenient to introduce a normal complex variable $a_k$
\begin{equation} \nonumber
 y_k =  \sqrt{\frac{\omega_k}{2g}}(a_k+a^*_{-k}) \hspace{.5cm}
\Pa_k =  -i\sqrt{\frac{2g}{\omega_k}}(a_k-a^*_{-k})
\end{equation}
\noindent which satisfies the equation of motion
\begin{equation} \nonumber
\frac{\partial a_k}{\partial t} + i\frac{\delta \Ha}{\delta a_k^*}=0.
\end{equation}
\noindent here $\omega_k = \sqrt{g|k|}$ -is the dispersion law for
the gravity waves.

In the normal variable $a_k$ second order term in the Hamiltonian
acquires the form:
\begin{eqnarray} \nonumber
\Ha_2 = \int \omega_k a_k a_k^* dk
\end{eqnarray}
\noindent The third order term is:
\begin{eqnarray} \nonumber
\Ha_3 & = &\frac{1}{2}\int\!V^{k_1}_{k_2 k_3}
\{a_{k_1}^* a_{k_2} a_{k_3}+a_{k_1} a_{k_2}^* a_{k_3}^*\}
\delta_{k_1-k_2-k_3}\!dk_1dk_2dk_3+\cr
&+&\frac{1}{6}\int\!U_{k_1 k_2 k_3}
\{a_{k_1} a_{k_2} a_{k_3} + a_{k_1}^* a_{k_2}^* a_{k_3}^*\}
\delta_{k_1+k_2+k_3}\!dk_1dk_2dk_3
\end{eqnarray}
\begin{eqnarray} \nonumber
V^{k_1}_{k_2 k_3} &=& \frac{g^{\frac{1}{4}}}{2\sqrt{4\pi}}
(|k_1|+|k_2|+|k_3|)\times\cr
&&\left(
|k_1 k_2 k_3|^{\frac{1}{4}} +
\left(k_1\over{k_2k_3}\right)^{1\over4}|k_1|-
\left(k_2\over{k_1k_3}\right)^{1\over4}|k_2|-
\left(k_3\over{k_1k_2}\right)^{1\over4}|k_3|
\right)\cr
U_{k_1 k_2 k_3} &=& \frac{g^{\frac{1}{4}}}{2\sqrt{4\pi}}
(|k_1|+|k_2|+|k_3|)\times\cr
&&\left(
|k_1 k_2 k_3|^{\frac{1}{4}} +
\left(k_1\over{k_2k_3}\right)^{1\over4}|k_1|+
\left(k_2\over{k_1k_3}\right)^{1\over4}|k_2|+
\left(k_3\over{k_1k_2}\right)^{1\over4}|k_3|
\right)
\end{eqnarray}
\noindent Fourth order term in the Hamiltonian consists of three terms:
\begin{eqnarray} \nonumber
\Ha_4 = \Ha_4^{4\Leftrightarrow 0} + \Ha_4^{3\Leftrightarrow 1} +
\Ha_4^{2\Leftrightarrow 2}
\end{eqnarray}
\noindent describing different types of the wave-wave interactions.
The term corresponding to $4\Leftrightarrow 0$
interaction has a form:
\begin{eqnarray} \nonumber
\Ha_4^{4\Leftrightarrow 0} = \frac{1}{24}\int &&R_{k_1 k_2 k_3 k_4}
(a_{k_1}a_{k_2}a_{k_3}a_{k_4}+a_{k_1}^*a_{k_2}^*a_{k_3}^*a_{k_4}^*)\times\cr
&&\delta_{k_1+k_2+k_3+k_4}\!dk_1dk_2dk_3dk_4
\end{eqnarray}

where $R_{k_1 k_2 k_3 k_4}$ is:

\begin{eqnarray} \nonumber
R_{k_1 k_2 k_3 k_4} = &\frac{-1}{4\pi}&(
{\left|\frac{k_1 k_2}{k_3 k_4}\right|}^\frac{1}{4}M^{k_1 k_2}_{k_3 k_4}+
{\left|\frac{k_1 k_3}{k_2 k_4}\right|}^\frac{1}{4}M^{k_1 k_3}_{k_2 k_4}+
{\left|\frac{k_1 k_4}{k_2 k_3}\right|}^\frac{1}{4}M^{k_1 k_4}_{k_2 k_3}+\cr
&+&{\left|\frac{k_2 k_3}{k_1 k_4}\right|}^\frac{1}{4}M^{k_2 k_3}_{k_1 k_4}+
{\left|\frac{k_2 k_4}{k_1 k_3}\right|}^\frac{1}{4}M^{k_2 k_4}_{k_1 k_3}+
{\left|\frac{k_3 k_4}{k_1 k_2}\right|}^\frac{1}{4}M^{k_3 k_4}_{k_1 k_2}
\end{eqnarray}

Term corresponding to $3\Leftrightarrow 1$
interaction has a form:
\begin{eqnarray} \nonumber
\Ha_4^{3\Leftrightarrow 1} = \frac{1}{6}\int &&G^{k_1}_{k_2 k_3 k_4}
(a_{k_1}^*a_{k_2}a_{k_3}a_{k_4}+a_{k_1}a_{k_2}^*a_{k_3}^*a_{k_4}^*)\times\cr
&&\delta_{k_1-k_2-k_3-k_4}\!dk_1dk_2dk_3dk_4
\end{eqnarray}

where $G^{k_1}_{k_2 k_3 k_4}$:

\begin{eqnarray} \nonumber
G^{k_1}_{k_2 k_3 k_4} = &\frac{-1}{4\pi}&(
{\left|\frac{k_1 k_2}{k_3 k_4}\right|}^\frac{1}{4}M^{-k_1 k_2}_{k_3 k_4}+
{\left|\frac{k_1 k_3}{k_2 k_4}\right|}^\frac{1}{4}M^{-k_1 k_3}_{k_2 k_4}+
{\left|\frac{k_1 k_4}{k_2 k_3}\right|}^\frac{1}{4}M^{-k_1 k_4}_{k_2 k_3}-\cr
&-&{\left|\frac{k_2 k_3}{k_1 k_4}\right|}^\frac{1}{4}M^{k_2 k_3}_{-k_1 k_4}-
{\left|\frac{k_2 k_4}{k_1 k_3}\right|}^\frac{1}{4}M^{k_2 k_4}_{-k_1 k_3}-
{\left|\frac{k_3 k_4}{k_1 k_2}\right|}^\frac{1}{4}M^{k_3 k_4}_{-k_1 k_2})
\end{eqnarray}

Term corresponding to $2\Leftrightarrow 2$
interaction has a form:
\begin{eqnarray} \nonumber
\Ha_4^{2\Leftrightarrow 2} = \frac{1}{4}\int W^{k_1 k_2}_{k_3 k_4}
a_{k_1}^*a_{k_2}^*a_{k_3}a_{k_4}
\delta_{k_1+k_2-k_3-k_4}dk_1dk_2dk_3dk_4
\end{eqnarray}


where $W^{k_1 k_2}_{k_3 k_4}$:

\begin{eqnarray} \nonumber
W^{k_1 k_2}_{k_3 k_4} = &\frac{-1}{4\pi}&(
{\left|\frac{k_1 k_2}{k_3 k_4}\right|}^\frac{1}{4}M^{-k_1 -k_2}_{k_3 k_4}-
{\left|\frac{k_1 k_3}{k_2 k_4}\right|}^\frac{1}{4}M^{-k_1 k_3}_{-k_2 k_4}-
{\left|\frac{k_1 k_4}{k_2 k_3}\right|}^\frac{1}{4}M^{-k_1 k_4}_{k_3 -k_2}-\cr
&-&{\left|\frac{k_2 k_3}{k_1 k_4}\right|}^\frac{1}{4}M^{-k_2 k_3}_{-k_1 k_4}-
{\left|\frac{k_2 k_4}{k_1 k_3}\right|}^\frac{1}{4}M^{-k_2 k_4}_{-k_1 k_3}+
{\left|\frac{k_3 k_4}{k_1 k_2}\right|}^\frac{1}{4}M^{k_3 k_4}_{-k_1 -k_2})
\end{eqnarray}

Among the different terms of the fifth order we consider only the term,
corresponding to the process (\ref{res5}):
\begin{eqnarray} \nonumber
\Ha_5 = \!\frac{1}{12}\int Q^{k_1 k_2 k_3}_{k_4 k_5}
\{a_{k_1}^* a_{k_2}^* a_{k_3}^* a_{k_4} a_{k_5}+\!c.c.\}
\delta_{k_1+k_2+k_3-k_4-k_5}\!dk_1dk_2dk_3dk_4dk_5
\end{eqnarray}
where

\begin{eqnarray} \nonumber
Q^{k_1 k_2 k_3}_{k_4 k_5}&=&
\frac{-3}{(4\pi)^{\frac{3}{2}}g^{\frac{1}{4}}}\times\{\cr
&+&({|k_1 k_2 k_3|\over{|k_4 k_5|}})^{1\over 4}N^{-k_1,-k_2,-k_3}_{k_4,k_5}-
({|k_1 k_2 k_4|\over{|k_3 k_5|}})^{1\over 4}N^{-k_1,-k_2,k_4}_{-k_3,k_5}-\cr
&-&({|k_1 k_2 k_5|\over{|k_3 k_4|}})^{1\over 4}N^{-k_1,-k_2,k_5}_{k_4,-k_3}-
({|k_1 k_3 k_4|\over{|k_2 k_5|}})^{1\over 4}N^{-k_1,k_4,-k_3}_{-k_2,k_5}-\cr
&-&({|k_1 k_3 k_5|\over{|k_2 k_4|}})^{1\over 4}N^{-k_1,k_5,-k_3}_{k_4,-k_2}-
({|k_2 k_3 k_4|\over{|k_1 k_5|}})^{1\over 4}N^{k_4,-k_2,-k_3}_{-k_1,k_5}-\cr
&-&({|k_2 k_3 k_5|\over{|k_1 k_4|}})^{1\over 4}N^{k_5,-k_2,-k_3}_{k_4,-k_1}+
({|k_1 k_4 k_5|\over{|k_2 k_3|}})^{1\over 4}N^{-k_1,k_4,k_5}_{-k_2,-k_3}+\cr
&+&({|k_2 k_4 k_5|\over{|k_1 k_3|}})^{1\over 4}N^{k_4,-k_2,k_5}_{-k_1,-k_3}+
({|k_3 k_4 k_5|\over{|k_1 k_2|}})^{1\over 4}N^{k_4,k_5,-k_3}_{-k_1,-k_2}\}
\end{eqnarray}

\section{Effective four-wave Hamiltonian}
\setcounter{equation}{0}

The Hamiltonian $\Ha$ in the normal variables $a_k$ is too complicated to
work with. Our purpose is to simplify the Hamiltonian to the form:
\begin{eqnarray} \label{Ham4_5}
\!&\Ha& = \int \omega_k b_k b_k^* dk + \frac{1}{4}\int T^{k_1 k_2}_{k_3 k_4}
b_{k_1}^*b_{k_2}^*b_{k_3}b_{k_4}
\delta_{k_1+k_2-k_3-k_4}dk_1dk_2dk_3dk_4+ \cr
%
\!&+&\!\!\frac{1}{12}\!\int\! T^{k_1 k_2 k_3}_{k_4 k_5}\!
\{b_{k_1}^* b_{k_2}^* b_{k_3}^* b_{k_4} b_{k_5}+\!c.c.\}
\delta_{k_1+k_2+k_3-k_4-k_5}\!dk_1dk_2dk_3dk_4dk_5
\end{eqnarray}
\noindent To do that we have to perform a transformation
(Zakharov, 1975 \cite{Z74}, Krasitskii, 1990 \cite{KRS90})
\begin{eqnarray} \label{transformation}
a_k &=& b_k + \int \Gamma^{k}_{k_1 k_2}b_{k_1}b_{k_2}\delta_{k-k_1-k_2} -
           2\int \Gamma^{k_2}_{k k_1}b^*_{k_1}b_{k_2}\delta_{k+k_1-k_2} +\cr
&+&\int \Gamma_{k k_1 k_2}b^*_{k_1}b^*_{k_2}\delta_{k+k_1+k_2} +
\int B^{k k_1}_{k_2 k_3}b^*_{k_1}b_{k_2}b_{k_3}\delta_{k+k_1-k_2-k_3}+\ldots
\end{eqnarray}
\begin{eqnarray} \nonumber
\Gamma^{k}_{k_1 k_2} = -\frac{1}{2}\frac{V^k_{k_1 k_2}}
                        {\omega_{k}-\omega_{k_1}-\omega_{k_2}} \hspace{0.3cm}
\Gamma_{k k_1 k_2} = -\frac{1}{2}\frac{U_{k k_1 k_2}}
                        {\omega_{k}+\omega_{k_1}+\omega_{k_2}}
\end{eqnarray}
\begin{eqnarray} \nonumber
B^{k k_1}_{k_2 k_3} &=& \Gamma^{k_1}_{k_2 k_1-k_2}\Gamma^{k_3}_{k k_3-k}+
\Gamma^{k_1}_{k_3 k_1-k_3}\Gamma^{k_2}_{k k_2-k}-\cr
&-&\Gamma^{k}_{k_2 k-k_2}\Gamma^{k_3}_{k_1 k_3-k_1}-
   \Gamma^{k_1}_{k_3 k_1-k_3}\Gamma^{k_2}_{k_1 k_2-k_1}-\cr
&-&\Gamma^{k+k_1}_{k k_1}\Gamma^{k_2+k_3}_{k_2 k_3}+
   \Gamma_{-k_1-k_1 k k_1}\Gamma_{-k_2-k_3 k_2 k_3}+
\tilde B^{k k_1}_{k_2 k_3}
\end{eqnarray}
\noindent Here $\tilde B^{k k_1}_{k_2 k_3}$ is an arbitrary function
satisfying the conditions:
\begin{eqnarray} \nonumber
\tilde B^{k k_1}_{k_2 k_3} = \tilde B^{k_1 k}_{k_2 k_3} =
\tilde B^{k k_1}_{k_3 k_2} = -(\tilde B^{k_2 k_3}_{k k_1})^*
\end{eqnarray}
\noindent The transformation (\ref{transformation}) is canonical up to
the terms of the order of $|b_k|^4$. It excludes from the Hamiltonian all
cubic terms. The form of the $T^{k k_1}_{k_2 k_3}$ depends on the choice
of the function $\tilde B^{k k_1}_{k_2 k_3}$. Let first
$\tilde B^{k k_1}_{k_2 k_3} = 0$.
Then $T^{k k_1}_{k_2 k_3} = \hat T^{k k_1}_{k_2 k_3}$ and
\begin{eqnarray} \label{T4_hat}
\hat T^{k k_1}_{k_2 k_3} = W^{k_1 k}_{k_2 k_3}- \hspace{9cm}\cr
-\frac{1}{2}V^{k}_{k_2 k-k_2}V^{k_3}_{k_1 k_3-k_1}
\left[\frac{1}{\omega_{k_2}+\omega_{k-k_2}-\omega_{k}}+
\frac{1}{\omega_{k_1}+\omega_{k_3-k_1}-\omega_{k_3}}\right]-\cr
-\frac{1}{2}V^{k_1}_{k_2 k_1-k_2}V^{k_3}_{k k_3-k}
\left[\frac{1}{\omega_{k_2}+\omega_{k_1-k_2}-\omega_{k_1}}
+\frac{1}{\omega_{k}+\omega_{k_3-k}-\omega_{k_3}}\right]-\cr
-\frac{1}{2}V^{k}_{k_3 k-k_3}V^{k_2}_{k_1 k_2-k_1}
\left[\frac{1}{\omega_{k_3}+\omega_{k-k_3}-\omega_{k}}
+\frac{1}{\omega_{k_1}+\omega_{k_2-k_1}-\omega_{k_2}}\right]-\cr
-\frac{1}{2}V^{k_1}_{k_3 k_1-k_3}V^{k_2}_{k k_2-k}
\left[\frac{1}{\omega_{k_3}+\omega_{k_1-k_3}-\omega_{k_1}}
+\frac{1}{\omega_{k}+\omega_{k_2-k}-\omega_{k_2}}\right]-\cr
-\frac{1}{2}V^{k+k_1}_{k k_1}V^{k_2+k_3}_{k_2 k_3}
\left[\frac{1}{\omega_{k+k_1}-\omega_{k}-\omega_{k_1}}
+\frac{1}{\omega_{k_2+k_3}-\omega_{k_2}-\omega_{k_3}}\right]-\cr
-\frac{1}{2}U_{-k-k_1 k k_1}U_{-k_2-k_3 k_2 k_3}
\left[\frac{1}{\omega_{k+k_1}+\omega_{k}+\omega_{k_1}}
+\frac{1}{\omega_{k_2+k_3}+\omega_{k_2}+\omega_{k_3}}\right]
\end{eqnarray}
\noindent The expression (\ref{T4_hat}) in spite of its complexity has
some remarkable properties. Let us consider all nontrivial solutions of the
equations (\ref{res4}). They consist of the manifold (\ref{nontriv4}) and
of seven other manifolds which are obtained from (\ref{nontriv4})  by
the permutations
\begin{eqnarray} \nonumber
k \leftrightarrow k_1, \hspace{.5cm}
k_2 \leftrightarrow k_3, \hspace{.5cm}
(k, k_1) \leftrightarrow (k_2, k_3)
\end{eqnarray}
Direct analytic calculation shows that
\[\hat T^{k k_1}_{k_2 k_3}\equiv 0\]
on all these manifolds. Recently Craig and Worfolk \cite{CW94} confirmed this
cancellation by the independent calculation. Another remarkable feature of
$\hat T^{k k_1}_{k_2 k_3}$ is the simplicity of its diagonal part. Let us
denote
\begin{eqnarray} \nonumber
T_{k k_1}  = \hat T^{k k_1}_{k k_1}
\end{eqnarray}
\noindent A simple, but long calculation (\cite{DZ94}, \cite{LZ}) shows that
\begin{eqnarray} \nonumber
T_{k k_1}  = \frac{1}{4\pi^2} k k_1 min\{|k|,|k_1|\}
\end{eqnarray}
\noindent Let us consider the function
\begin{eqnarray} \nonumber
\tilde T^{k k_1}_{k_2 k_3}  &=& [
\frac{1}{2}(T_{k k_2} + T_{k k_3} + T_{k_1 k_2} + T_{k_1 k_3}) -\cr
&-&\frac{1}{4}(T_{k k} + T_{k_1 k_1} + T_{k_2 k_2} + T_{k_3 k_3})]
\theta(k k_1 k_2 k_3)
\end{eqnarray}
\[\theta(x) = \cases{1,&if $x\ge0$;\cr
                       0,&if $x<0.$}\]
\noindent Obviously
\begin{eqnarray} \nonumber
\tilde T^{k k_1}_{k k_1} = \hat T^{k k_1}_{k k_1} = T_{k k_1}
\end{eqnarray}
\noindent Let us choose
\begin{eqnarray} \nonumber
\tilde B^{k k_1}_{k_2 k_3} =
\frac{\tilde T^{k k_1}_{k_2 k_3} - \hat T^{k k_1}_{k_2 k_3}}
{\omega_{k} + \omega_{k_1} - \omega_{k_2} - \omega_{k_3}}
\end{eqnarray}
\noindent One can check that this transformation makes replacement
of $\hat T^{k k_1}_{k_2 k_3}$ by $\tilde T^{k k_1}_{k_2 k_3}$. So,
one can assume in future that
\begin{eqnarray} \nonumber
T^{k k_1}_{k_2 k_3} = \tilde T^{k k_1}_{k_2 k_3}
\end{eqnarray}
In the case of periodic boundary conditions
\begin{eqnarray} \nonumber
b_k = \sum_{n=-\infty}^{\infty} {b_n \delta_{k-k_0n}}
\end{eqnarray}
\noindent Introducing the notation
\begin{eqnarray} \nonumber
\omega_{k_0n} &=& \omega_n \cr
T^{k_0n k_0n_1}_{k_0n_2 k_0n_3} &=& T^{n n_1}_{n_2 n_3}
\end{eqnarray}
\noindent we can find that now
\begin{eqnarray} \nonumber
\frac{\partial b_n}{\partial t} + i\frac{\delta \Ha}{\delta b_n^*}=0.
\end{eqnarray}
\begin{eqnarray} \nonumber
\Ha &=& \sum_{n=-\infty}^{\infty} \omega_n |b_n|^2 +
\frac{1}{4}\sum_{n,n_1,n_2,n_3=-\infty}^{\infty} T^{n n_1}_{n_2 n_3}
b_{n}^*b_{n_1}^*b_{n_2}b_{n_3}
\delta_{n+n_1,n_2+n_3}
\end{eqnarray}
\noindent In the canonical transformation (\ref{transformation}) all the
integrals are replaced now by the discrete sums. In particulary instead of
$\tilde B^{k k_1}_{k_2 k_3}$ we have now
\[\tilde B^{n n_1}_{n_2 n_3} = \tilde B^{k_0n k_0n_1}_{k_0n_2 k_0n_3}\]
\noindent Let us choose
\begin{eqnarray} \label{Bntilde}
\tilde B^{n n_1}_{n_2 n_3} = \frac
{T^{n n_1}_{n_2 n_3} - T_{n n_1}\delta_{n,n_2}\delta_{n_1,n_3}
- T_{n n_1}\delta_{n,n_3}\delta_{n_1,n_2}}
{\omega_{n} + \omega_{n_1} - \omega_{n_2} - \omega_{n_3}}
\end{eqnarray}
\noindent This expression has no singularities on the diagonals
$n_2=n, n_3=n_1$ and $n_2=n_1, n_3=n$. The transformation
(\ref{transformation}) with (\ref{Bntilde}) brings the Hamiltonian to
the Birghoff's form
\begin{eqnarray} \label{Birghoff}
\Ha &=& \sum_{n=-\infty}^{\infty} \omega_n |b_n|^2 +
\frac{1}{4}\sum_{n,n_1=-\infty}^{\infty} T_{n n_1}
|b_{n}|^2|b_{n_1}|^2
\end{eqnarray}
\noindent This is a Hamiltonian of an integrable system. A level of
nonlinearity, allowing the representation (\ref{Birghoff}) has to be studied
separately.

\section{Five-wave interaction on resonant surface}
\setcounter{equation}{0}

To find $T^{k k_1 k_2}_{k_3 k_4}$ one can calculate the terms of the
order of $b^3$ and $b^4$ in the canonical transformation
(\ref{transformation}). This very cumbersome procedure was fulfilled
by V.Krasitskii\cite{KRS94}. This is a kind of feat, but the resulting
formulae are so complicated that hardly can be used for any practical
purpose. In this article we offer much more simple and clear way to
$T^{k k_1 k_2}_{k_3 k_4}$ involving a Feinman's diagram technique for
the scattering matrix.

Our intermediate formulae are very complicated also, due to one-to-one
correspondense of each term in the expression to a certain graphic
picture, however all the procedure is much easy controlled. It is
very remarkable that our final formula is very simple.

First we introduce so called formal classical scattering matrix. Let
\begin{eqnarray} \nonumber
H = H_2 + H_{int}
\end{eqnarray}
\noindent is a Hamiltonian of a some nonlinear system in a homogeneous
space. Here $H_2 = \int \omega_k a_k a^*_k dk$, and $H_{int}$ is in
general case an infinite series in power $a_k, a^*_k$
\begin{eqnarray} \nonumber
H_{int} = H_3 + H_4 + \ldots
\end{eqnarray}
\noindent The motion equation is as usual
\begin{equation} \label{motionS}
\frac{\partial a_k}{\partial t} + i\frac{\delta H}{\delta a_k^*}=0.
\end{equation}
\noindent One can change $H$ to the auxiliary Hamiltonian
\begin{equation} \nonumber
\hat H = H_2 + e^{-\epsilon |t|} H_{int}
\end{equation}
\noindent Now the equation (\ref{motionS}) becomes linear at
$t \rightarrow \pm \infty$ and
\begin{eqnarray} \nonumber
a_k(t) &\rightarrow&  c^{\pm}_ke^{-i\omega_k t} \cr
t &\rightarrow&  \pm \infty
\end{eqnarray}
\noindent The asymptotic states $c^{\pm}_k$ are not independent,
and actually
\begin{eqnarray} \nonumber
c^{+}_k = \hat S_{\epsilon}[c^{-}_k]
\end{eqnarray}
\noindent $\hat S_{\epsilon}[c^{-}_k]$ is a nonlinear operator
which can be presented as a series in power of $c^{-}, {c^{-}}^{*}$.
We will treat this series as formal one and will not care about their
convergence. A formal series which is a result of the limiting
transition
\begin{eqnarray} \nonumber
\hat S[c^{-}_k] = \lim_{\epsilon\to0}{\hat S_{\epsilon}[c^{-}_k]}
\end{eqnarray}
\noindent is the formal classic scattering matrix. It has the following
form
\begin{eqnarray} \label{ScMat}
\hat S[c^{-}_k] = c^{-}_k &-& \sum_{n+m\ge3}^{}
\frac{2\pi i}{(n-1)!m!}\int
S_{n m}(k,k_1,\ldots,k_{n-1};k_n,\ldots,k_{n+m-1})\times\cr
&\times&\delta_{k+k_1+\ldots +k_{n-1}-k_n-\ldots -k_{n+m-1}}
\delta_{\omega_k+\omega_{k_1}+\ldots +\omega_{k_{n-1}}
-\omega_{k_n}-\ldots -\omega_{k_{n+m-1}}}\times\cr
&\times&{c^-}^*_{k_1}\ldots {c^-}^*_{k_{n-1}}c^-_{k_n}\ldots c^-_{k_{n+m-1}}
dk_1\ldots dk_{n+m-1}
\end{eqnarray}
The functions $S_{n m}$ are the elements of the scattering matrix. They
are defined on the resonant manifolds
\begin{eqnarray} \label{resN}
k+k_1+\ldots +k_{n-1} &=&k_n+\ldots +k_{n+m-1}\cr
\omega_k+\omega_{k_1}+\ldots +\omega_{k_{n-1}} &=&
\omega_{k_n}+\ldots +\omega_{k_{n+m-1}}
\end{eqnarray}
Two basic properties of the matrix elemets are important for us.
\begin{enumerate}
\item{The value of the matrix element $S_{n m}$ on the resonant manifold
(\ref{resN}) is invariant with respect to canonical transformation
(\ref{transformation}).}
\item{There is a simple algorythm for calculation of the matrix elements.
The element $S_{n m}$ is a finite sum of terms which can be expressed
through the coefficients of the Hamiltonians $H_i, i\le {n+m}$. Each term
can be marked by a certain Feinman's diagram, having no internal loops.
The rules of correspondense are described in the Appendix B.}
\end{enumerate}
Actually the classical scattering matrix is nothing but the Feinman's
scattering matrix taken in the 'tree' approximation. This approximation
makes a number of terms being finite for any element.

Our idea how to find $T^{k k_1 k_2}_{k_3 k_4}$ is the follow. We
calculate first nonzero elements of the scattering matrix for
the Hamiltonian (\ref{HamSer}) and for the Hamiltonian (\ref{Ham4_5}).
Because of these two Hamiltonians are connected by the canonical
transformation (\ref{transformation}), the results must coinside.
For surface gravity waves first nontrivial matrix element is $S_{2 2}$.
In terms of the Hamiltonian (\ref{Ham4_5}) it is
\begin{eqnarray} \nonumber
S_{2 2}(k,k_1,k_2,k_3) = T^{k k_1}_{k_2 k_3}
\end{eqnarray}
\noindent Being calculated for the Hamiltinian (\ref{HamSer}), this
element consist of six terms. They are presented (together with
corresponding diagrams) in the Appendix B. One can see that the result
coinside with the expression (\ref{T4_hat}) on the resonant manifold
(\ref{res4}).

In one-dimensional case the first integral in (\ref{ScMat}) can be
calculated so that the first two terms in (\ref{ScMat}) has a form
\begin{eqnarray} \label{c2c}
c^+_k = c^-_k (1 - \pi i\int_{-\infty}^{\infty}
\frac{T_{k k_1}}{|\omega_k - \omega_{k_1}|}|c^-_{k_1}|^2 dk_1)
\end{eqnarray}
\noindent This formula one more time stresses the fact that four-wave
nonlinear processes in one-dimensional case lead only to the trivial
scattering which does not produce ``new wave vectors''. The integral
in (\ref{c2c}) diverges logarithmically. It is why our scattering matrix
is ``formal''. In reality in the one dimensional case the waves don't
became a linear indeed if $t\to \infty$. They acquire logarithmically
growing phase (see Zakharov, Manakov \cite{ZM}).

The first nontrivial element of the scattering matrix in the
one-dimensional case is
\begin{eqnarray} \nonumber
S_{3 2}(k,k_1,k_2,k_3,k_4) = T^{k k_1 k_2}_{k_3 k_4}
\end{eqnarray}
\noindent Being calculated in terms of the initial Hamiltonian (\ref{HamSer})
it consists of 81 terms. Their expressions together with diagrams are
presented in the Appendix C.

In spite of complexity of the expression for $T^{k k_1 k_2}_{k_3 k_4}$
it can be enormousely simplified on the resonant manifold. We will
discuss here only the case when all $k_i$ in the resonat conditions
\begin{eqnarray} \label{res5a}
k + k_1 + k_2 &=& k_3 + k_4 \cr
\omega_{k} + \omega_{k_1} + \omega_{k_2} &=& \omega_{k_3} + \omega_{k_4}
\end{eqnarray}
\noindent have the same sign.

The manifold (\ref{res5a}) can be parametrised as follow
\begin{eqnarray} \label{param}
k   & = & a (p^2 - q^2 +1 - 2p)^2,\cr
k_1 & = & a (p^2 - q^2 +1 + 2p)^2,\cr
k_2 & = & 16 a, \cr
k_3 & = & a (p^2 - q^2 +3 - 2q)^2,\cr
k_4 & = & a (p^2 - q^2 +3 + 2q)^2,
\end{eqnarray}
\noindent here $0 < |p|,|q| < 1, |p\pm q| < 1, a > 0$. Easy to see
that $k_i$ here satisfy the inequality:
\begin{eqnarray} \nonumber
k, k_1 < k_3, k_4 < k_2
\end{eqnarray}
\noindent Plugging the parametrization (\ref{param}) in the expression
obtained for $T^{k k_1 k_2}_{k_3 k_4}$ we get a sum of more than thousands
terms. Using the program for analytical calculations 'Mathematica' we
manage to simplify this expression to the following form
\begin{eqnarray} \label{T5p}
T^{k_1 k_2 k_3}_{k_4 k_5} = \frac{2}{g^{1/2}\pi^{3/2}}
\sqrt{\frac{\omega_{k_1} \omega_{k_2} \omega_{k_3}}{\omega_{k_4} \omega_{k_5}}}
\frac{k_1 k_2 k_3 k_4 k_5}{max(k_1,k_2,k_3)}
\end{eqnarray}
This formula is the main result of the presented article. The fact that
$T^{k k_1 k_2}_{k_3 k_4}\neq0$ on the resonant surface means that the system
of gravity waves on a surface of deep water is \underline{nonintegrable
Hamiltonian system}.


\section{Five-wave kinetic equation and its solutions}
\setcounter{equation}{0}
The matrix element $T^{k k_1}_{k_2 k_3 k_4}$ for
$3\Leftrightarrow 2$ process along with
$T^{k k_1}_{k_2 k_3}$
allows us to derive kinetic equation which includes four- and five-wave
interactions.

The dynamical equation for $b_k$ (with the Hamiltonian (\ref{Ham4_5})) is:
\begin{eqnarray} \label{eq:dynamic}
\frac{\partial b_k}{\partial t}&+&i\omega_k b_k +
i\frac{1}{2}\int T^{k k_1}_{k_2 k_3}
b^*_{k_1} b_{k_2} b_{k_3} \delta_{k+k_1-k_2-k_3}dk_1dk_2dk_3 + \cr
 &+& i\frac{1}{4}\int T^{k k_2 k_3}_{k_4 k_5}
b^*_{k_2}b^*_{k_3}b_{k_4}b_{k_5}\delta_{k+k_2+k_3-k_4-k_5}dk_2dk_3dk_4dk_5+\cr
 &+& i\frac{1}{6}\int T^{k_1 k_2 k_3}_{k k_5}
b_{k_1}b_{k_2}b_{k_3}b^*_{k_5}\delta_{k_1+k_2+k_3-k-k_5}dk_1dk_2dk_3dk_5 = 0.
\hspace{0.8cm}
\end{eqnarray}
\noindent Introducing standard pair correlation function $n_k$
\begin{eqnarray} \nonumber
<b_k b_{k_1}> = n_k \delta_{k-k_1}
\end{eqnarray}
\noindent
we can derive from (\ref{eq:dynamic}) the equation for $n_k$:
\begin{eqnarray} \nonumber
\frac{\partial n_k}{\partial t}&=& {\tt Im} \int
T^{k k_1}_{k_2 k_3}<b^*_{k}b^*_{k_1}b_{k_2}b_{k_3}>
\delta_{k+k_1-k_2-k_4}dk_1dk_2dk_3 + \cr
&+&\frac{1}{2}{\tt Im}\int
T^{k k_2 k_3}_{k_4 k_5}<b^*_{k}b^*_{k_2}b^*_{k_3}b_{k_4}b_{k_5}>
\delta_{k+k_2+k_3-k_4-k_5}dk_2dk_3dk_4dk_5 - \cr
&-&\frac{1}{3}{\tt Im}\int
T^{k_1 k_2 k_3}_{k k_5}<b^*_{k_1}b^*_{k_2}b^*_{k_3}b_{k}b_{k_5}>
\delta_{k_1+k_2+k_3-k-k_5}dk_1dk_2dk_3dkdk_5
\end{eqnarray}
\noindent It is obvious that $T^{k k_1}_{k_2 k_3}$ contributes to the
equation for fourth-order correlator, while $T^{k_1 k_2 k_3}_{k_4 k_5}$
contributes to the equation for fifth-order correlator only (due to
the seventh-order correlators vanish).
Fifth-order correlation function
$<b^*_{k_1}b^*_{k_2}b^*_{k_3}b_{k_4}b_{k_5}>$  can be expressed through
the eighth-order correlator:
\begin{eqnarray} \nonumber
&&(\frac{\partial }{\partial t}-
i(\omega_{k_1}+\omega_{k_2}+\omega_{k_3}-\omega_{k_4}-\omega_{k_5}))
<b^*_{k_1}b^*_{k_2}b^*_{k_3}b_{k_4}b_{k_5}> = \cr
&=&i\frac{1}{4}\int T^{k_1 p_2 p_3}_{p_4 p_5}
<b^*_{k_2}b^*_{k_3}b^*_{p_4}b^*_{p_5}b_{k_4}b_{k_5}b_{p_2}b_{p_3}>
\delta_{k_1+p_2+p_3-p_4-p_5}dp_2dp_3dp_4dp_5 + \cr
&+&i\frac{1}{4}\int T^{k_2 p_2 p_3}_{p_4 p_5}
<b^*_{k_1}b^*_{k_3}b^*_{p_4}b^*_{p_5}b_{k_4}b_{k_5}b_{p_2}b_{p_3}>
\delta_{k_2+p_2+p_3-p_4-p_5}dp_2dp_3dp_4dp_5 + \cr
&+&i\frac{1}{4}\int T^{k_3 p_2 p_3}_{p_4 p_5}
<b^*_{k_1}b^*_{k_2}b^*_{p_4}b^*_{p_5}b_{k_4}b_{k_5}b_{p_2}b_{p_3}>
\delta_{k_3+p_2+p_3-p_4-p_5}dp_2dp_3dp_4dp_5 -\cr
&-&i\frac{1}{6}\int T^{p_1 p_2 p_3}_{k_4 p_5}
<b^*_{k_1}b^*_{k_2}b^*_{k_3}b^*_{p_5}b_{k_5}b_{p_1}b_{p_2}b_{p_3}>
\delta_{p_1+p_2+p_3-k_4-p_5}dp_1dp_2dp_3dp_5 -\cr
&-&i\frac{1}{6}\int T^{p_1 p_2 p_3}_{k_5 p_5}
<b^*_{k_1}b^*_{k_2}b^*_{k_3}b^*_{p_5}b_{k_4}b_{p_1}b_{p_2}b_{p_3}>
\delta_{p_1+p_2+p_3-k_5-p_5}dp_1dp_2dp_3dp_5
\end{eqnarray}
\noindent Applying random phase approximation for the eighth-order correlator
(to split it in a product of pair correlation functions) and assuming
slow variation in time for fifth-order correlator, one can
get the following expression for
$<b^*_{k_1}b^*_{k_2}b^*_{k_3}b_{k_4}b_{k_5}>$:
\begin{eqnarray} \label{eq:5cor}
&&{\tt Im}<b^*_{k_1}b^*_{k_2}b^*_{k_3}b_{k_4}b_{k_5}> =
\pi T^{k_1 k_2 k_3}_{k_4 k_5}
\delta_{\omega_{k_1}+\omega_{k_2}+\omega_{k_3}-
\omega_{k_4}-\omega_{k_5}}\times \cr
&\times& \{ n_{k_1}n_{k_2}n_{k_3}(n_{k_4}+n_{k_5}) -
n_{k_4}n_{k_5}(n_{k_1}n_{k_2}+n_{k_1}n_{k_3}+n_{k_2}n_{k_3})\}
\end{eqnarray}
\noindent In the (\ref{eq:5cor}) we drop the terms which are out of resonant
surface. For fourth-order correlator we have the following
equaton \cite{ZLF92}:
\begin{eqnarray} \label{eq:4cor}
&&{\tt Im}<b^*_{k}b^*_{k_1}b_{k_2}b_{k_3}> =
\pi T^{k k_1}_{k_2 k_3}
\delta_{\omega_{k}+\omega_{k_1}-\omega_{k_2}-\omega_{k_3}}\times \cr
&\times& \{ n_{k_2}n_{k_3}(n_{k}+n_{k_1}) -
n_{k}n_{k_1}(n_{k_2}+n_{k_3})\}
\end{eqnarray}
\noindent Substituting (\ref{eq:4cor}) and (\ref{eq:5cor}) into the
equation for $n_k$
we get five-wave kinetic equation:
\begin{eqnarray} \nonumber
\frac{\partial n_k}{\partial t} &=& st(n,n,n) + st(n,n,n,n)
\end{eqnarray}
\begin{eqnarray} \nonumber
st(n,n,n) &=&\pi\int
|T^{k k_1}_{k_2 k_3}|^2
f^{k k_1}_{k_2 k_3}dk_1dk_2dk_3\cr
st(n,n,n,n) &=&\frac{\pi}{3}\int
|T^{k_1 k_2 k_3}_{k k_5}|^2
f^{k_1 k_2 k_3}_{k k_5}dk_1dk_2dk_3dk_5 - \cr
&-&\frac{\pi}{2}\int
|T^{k k_2 k_3}_{k_4 k_5}|^2
f^{k k_2 k_3}_{k_4 k_5}dk_2dk_3dk_4dk_5,
\end{eqnarray}
\begin{eqnarray} \nonumber
f^{k k_1}_{k_2 k_3} &=& \delta_{k+k_1-k_2-k_3}
\delta_{\omega_{k}+\omega_{k_1}
-\omega_{k_2}-\omega_{k_3}}\times \cr
&\times& \{ n_{k_2}n_{k_3}(n_{k}+n_{k_1}) -
n_{k}n_{k_1}(n_{k_2}+n_{k_3})\}\cr
f^{k_1 k_2 k_3}_{k_4 k_5} &=& \delta_{k_1+k_2+k_3-k_4-k_5}
\delta_{\omega_{k_1}+\omega_{k_2}+\omega_{k_3}
-\omega_{k_4}-\omega_{k_5}}\times \cr
&\times& \{ n_{k_1}n_{k_2}n_{k_3}(n_{k_4}+n_{k_5}) -
n_{k_4}n_{k_5}(n_{k_1}n_{k_2}+n_{k_1}n_{k_3}+n_{k_2}n_{k_3})\}
\end{eqnarray}
As it was shown in the Introction, In the one dimensional case
$st(n,n,n)\equiv0$ and we end up with the pure five-wave kinetic
equation:
\begin{eqnarray} \label{eq:kinetic}
\frac{\partial n_k}{\partial t} &=&st(n,n,n,n)
\end{eqnarray}
The equation (\ref{eq:kinetic}) formally preserves two integrals of motion,
energy
\begin{eqnarray} \nonumber
E = \int_{0}^{\infty} \omega_{k}n_k dk,
\end{eqnarray}
\noindent  and momentum
\begin{eqnarray} \nonumber
P = \int_{0}^{\infty} k n_k dk
\end{eqnarray}
\noindent (We consider the case when all $k_i$ are positive).
The stationary equation
\begin{eqnarray} \label{stat5}
st(n,n,n,n) = 0
\end{eqnarray}
\noindent has thermodynamic solution
\begin{eqnarray} \nonumber
n_k = \frac{T}{\omega_{k}+\alpha k}.
\end{eqnarray}
\noindent Like four-wave isotropic kinetic equation, the equation
(\ref{eq:kinetic}) describes direct and inverse cascades. The inverse
cascade is the cascade of energy, which is a real constant of motion
and is carried toward the small $k$. It is described by the following
Kolmogorov solution of the equation (\ref{stat5})
\begin{eqnarray} \nonumber
n^{(1)}_k = \alpha^{(1)} \epsilon^{1/4} |k|^{-25/8}
\end{eqnarray}
\noindent Here $\epsilon$ is the energy flux, $\alpha^{(1)}$ is the
Kolmogorov constant.

A corresponding energy spectrum is
\begin{eqnarray} \nonumber
\cal E_{\omega}d\omega &=& \omega_k n_k dk\cr
\cal E_{\omega} &=& \alpha^{(1)} \epsilon^{1/4}{\omega}^{-17/4}
\end{eqnarray}
Direct cascade is a transport of momentum towards the large wave numbers.
It is described by the Kolmogorov solution
\begin{eqnarray} \nonumber
n^{(2)}_k = \alpha^{(2)} \mu^{1/4} |k|^{-13/4}
\end{eqnarray}
\noindent Here $\mu$ is the momentum flux, $\alpha^{(2)}$ is the Kolmogorov
constant.

Now
\begin{eqnarray} \nonumber
\cal E_{\omega} &=& \alpha^{(2)} \mu^{1/4}{\omega}^{-9/2}
\end{eqnarray}
Due to the direct cascade the momentum is not a real constant of motion,
it leaks permanently to the large $k$ region. More detailed description
of the Kolmogorov spectra in one-dimensional case will be published
separately.

In conclusion authors express gratitude to Dr. Victor L'vov for his
valuable advises and to Alina Spectorov for  preparing the diagrams.

This work is supported by the ONR Grant N00 14--92--J--1343 and Russian
Basic Research Foundation Grant N00 94-01-00898. Y.L. acknowledges
support from AFOSR Grant F49620-94-1-0144DEF.


\appendix
\section{Appendix}
\renewcommand{\theequation}{\thesubsection.\arabic{equation}}
\setcounter{equation}{0}

\subsection{4th and 5th-order terms of the Hamiltonian}

First part of $\Ha_4$ is
\begin{eqnarray}
{\Ha_4}^{1st} &=& -{1\over 2} \int 2\Ps^{(3)} H{\Ps^{(1)}}^\prime du =\cr
 &=&-\int \Pa ({x^\prime}^2-{y^\prime}^2) H{\Pa^\prime} du
+2\int x^\prime H(y^\prime \Pa) H{\Pa^\prime} du = \cr
&=&\frac{1}{2\pi}\int {|k_3|(|k_1 k_2|+k_1 k_2)
z_{k_1} z_{k_2} \Pa_{k_3} \Pa_{k_4}
\delta_{k_1+k_2+k_3+k_4} dk_1 dk_2 dk_3 dk_4} +\cr
&+&\frac{1}{\pi}\int {|k_1| k_2 |k_3|s_{k_2+k_4}
z_{k_1} z_{k_2} \Pa_{k_3} \Pa_{k_4}
\delta_{k_1+k_2+k_3+k_4} dk_1 dk_2 dk_3 dk_4} \nonumber
\end{eqnarray}

Second part of $\Ha_4$ is
\begin{eqnarray} \nonumber
{\Ha_4}^{2nd} &=& -{1\over 2} \int \Ps^{(2)} H{\Ps^{(2)}}^\prime du =\cr
&=&-{1\over 2} \int (y^\prime\Pa) H(y^\prime \Pa)^\prime +
(x^\prime\Pa) H(x^\prime \Pa)^\prime
+2 (x^\prime \Pa) (y^\prime \Pa)^\prime du
\end{eqnarray}
In the $k$-space it acquires the form:
\begin{eqnarray}
{\Ha_4}^{2nd} = &\frac{1}{4\pi}&\int
\{(|k_1 k_2|-k_1 k_2) |k_2+k_4| + (|k_1| k_2 - k_1 |k_2|) (k_2+k_4)\}\times\cr
&&z_{k_1} z_{k_2} \Pa_{k_3} \Pa_{k_4}
\delta_{k_1+k_2+k_3+k_4} dk_1 dk_2 dk_3 dk_4
\end{eqnarray}

First part of $\Ha_5$ is
\begin{eqnarray} \nonumber
{\Ha_5}^{1st} = -&\int& \Ps^{(4)} H{\Ps^{(1)}}^\prime du =\cr
\frac{-1}{(2\pi)^{3\over 2}}&&\int|k_4|\{(3k_1 k_2 +|k_1 k_2|)|k_3| +
           (3|k_1 k_2|+k_1 k_2)s_{k_3+k_5} k_3\}\times\cr
&&z_{k_1}z_{k_2} z_{k_3} \Pa_{k_4} \Pa_{k_5}
\delta_{k_1+k_2+k_3+k_4+k_5} dk_1 dk_2 dk_3 dk_4 dk_5
\end{eqnarray}

Second part of $\Ha_5$ is
\begin{eqnarray}
{\Ha_5}^{2nd} = -&\int& \Ps^{(3)} H{\Ps^{(2)}}^\prime du =\cr
&\int& \Pa ({x^\prime}^2-{y^\prime}^2)(z^\prime\Pa)^\prime du +
\int \Pa ({x^\prime}^2-{z^\prime}^2) H(x^\prime\Pa)^\prime du -\cr
-2&\int& x^\prime H(z^\prime \Pa)(z^\prime\Pa^\prime)^\prime du
-2\int x^\prime H(z^\prime \Pa) H(x^\prime\Pa)^\prime du = \cr
-\frac{1}{(2\pi)^{3\over 2}}&\int&\!\!\{(|k_1 k_2|+k_1 k_2)(k_3(k_3+k_4)+|k_3||k_3+k_4|)\}\times\cr
&&z_{k_1} z_{k_2} z_{k_3}\Pa_{k_4} \Pa_{k_5}
\delta_{k_1+k_2+k_3+k_4+k_5} dk_1 dk_2 dk_3 dk_4 dk_5 -\cr
-\frac{2}{(2\pi)^{3\over 2}}&\int& |k_1| k_3 s_{k_3+k_4}
(k_2 (k_2 + k_5) + |k_2||k_2 + k_5|) \times\cr
&&z_{k_1} z_{k_2} z_{k_3}\Pa_{k_4} \Pa_{k_5}
\delta_{k_1+k_2+k_3+k_4+k_5} dk_1 dk_2 dk_3 dk_4 dk_5 \nonumber
\end{eqnarray}

Hamiltonian can be written in symmetrical form:
\begin{eqnarray} \nonumber
\Ha = \frac{1}{2}&\int&\!(g |z_k|^2 + |k||\Pa_k|^2) dk +\cr
+\frac{1}{2\sqrt{2\pi}}&\int&\!\{S_{k_1 k_2 k_3}z_{k_1}z_{k_2}z_{k_3} -
F^{k_1}_{k_2 k_3}z_{k_1}\Pa_{k_2} \Pa_{k_3}\}
 \delta_{k_1+k_2+k_3}dk_1dk_2dk_3 +\cr
+\frac{1}{4\pi}&\int&\!M^{k_1 k_2}_{k_3 k_4}z_{k_1}z_{k_2}\Pa_{k_3}\Pa_{k_4}
\delta_{k_1+k_2+k_3+k_4}dk_1dk_2dk_3dk_4 +\cr
+\frac{1}{2(2\pi)^{3\over 2}}&\int&\!N^{k_1 k_2 k_3}_{k_4 k_5}
z_{k_1}z_{k_2}z_{k_3}\Pa_{k_4}\Pa_{k_5}
\delta_{k_1+k_2+k_3+k_4+k_5}dk_1dk_2dk_3dk_4dk_5
\end{eqnarray}
\noindent where $S_{k_1 k_2 k_3}$, $L_{k_1 k_2}$ and $F^{k_1}_{k_2 k_3}$
are defined in (\ref{SLF}), and
\begin{eqnarray}
M^{k_1 k_2}_{k_3 k_4} &=& (|k_3|+|k_4|)(k_1 k_2 +|k_1 k_2|) +\cr
&+&{1\over 4}L_{-k_1 k_2}(|k_1+k_3|+|k_1+k_4|+|k_2+k_3|+|k_2+k_4|)+\cr
&+&{1\over 2}(|k_1|k_2(2k_2+k_3+k_4)+k_1|k_2|(2k_1+k_3+k_4))+\cr
&+&|k_1|k_2(|k_3|s_{k_2+k_4}+|k_4|s_{k_2+k_3})+
k_1|k_2|(|k_3|s_{k_1+k_4}+|k_4|s_{k_1+k_3})
\nonumber
\end{eqnarray}
\begin{eqnarray}
N^{k_1 k_2 k_3}_{k_4 k_5} &=&\cr
&-&\{(|k_1 k_2 k_3| + |k_1| k_2 k_3 + k_1 |k_2| k_3 + k_1 k_2 |k_3|)
(|k_4| + |k_5|) + \cr
&+&\frac{k_1 k_2 k_3}{3}(|k_4|(s_{k_1+k_5}+s_{k_2+k_5}+s_{k_3+k_5})+\cr
&+&|k_5|(s_{k_1+k_4}+s_{k_2+k_4}+s_{k_3+k_4}))+\cr
&+&|k_4|(k_1|k_2||k_3| s_{k_1+k_5}+
       |k_1|k_2|k_3| s_{k_2+k_5}+|k_1||k_2| k_3 s_{k_3+k_5})+\cr
&+&|k_5|(k_1|k_2||k_3| s_{k_1+k_4}+
       |k_1|k_2|k_3| s_{k_2+k_4}+|k_1||k_2| k_3 s_{k_3+k_4})\}\cr
\cr
&-&\frac{1}{3}\{
L_{k_1,k_2} L_{k_3,k_3+k_4} + L_{k_3,k_2} L_{k_1,k_1+k_4}+
L_{k_1,k_3} L_{k_2,k_2+k_4}+\cr
&+&L_{k_1,k_2} L_{k_3,k_3+k_5} + L_{k_3,k_2} L_{k_1,k_1+k_5}+
L_{k_1,k_3} L_{k_2,k_2+k_5}
\}-\cr
-\frac{1}{3}\{
|k_1|&(&k_3(L_{k_2,k_2+k_4}s_{k_3+k_5}+ L_{k_2,k_2+k_5}s_{k_3+k_4})+\cr
     &+&k_2(L_{k_3,k_3+k_4}s_{k_2+k_5}+ L_{k_3,k_3+k_5}s_{k_2+k_4}))+\cr
%
|k_2|&(&k_3(L_{k_1,k_1+k_4}s_{k_3+k_5}+ L_{k_1,k_1+k_5}s_{k_3+k_4})+\cr
     &+&k_1(L_{k_3,k_3+k_4}s_{k_1+k_5}+ L_{k_3,k_3+k_5}s_{k_1+k_4}))+\cr
%
|k_3|&(&k_2(L_{k_1,k_1+k_4}s_{k_2+k_5}+ L_{k_1,k_1+k_5}s_{k_2+k_4})+\cr
     &+&k_1(L_{k_2,k_2+k_4}s_{k_1+k_5}+ L_{k_2,k_2+k_5}s_{k_1+k_4}))\}
\nonumber
\end{eqnarray}


\def \eeb {\begin{flushright}
          \vskip -0.5cm
          \begin{minipage}{9cm}}
\def \eep {\end{minipage}
         \end{flushright}}

\def \eb {\begin{flushright}
          \vskip -1.2cm
          \begin{minipage}{9cm}}
\def \ep {\end{minipage}
         \end{flushright}}

%\newpage
\section{Appendix}

\setlength{\unitlength}{.08cm}
 Let  us introduce basic object of diagrammatic technique we use in this
work.

\begin{enumerate}
\item
Bare fourth order vertex with 2 incoming and 2 outgoing wave vectors:

\begin{picture}(36,30)(0,10)
\put(0,24){\vector(1,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\multiput(13,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(13,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(13,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(13,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(23.1,23){\vector(1,1){1}}
\put(23.1,1){\vector(1,-1){1}}


\put(4,0){2}
\put(3,24){1}
\put(19,24){3}
\put(18,0){4}
\end{picture}

\eeb
\[ W^{k_1 k_2}_{k_3 k_4}\]
   \eep

\vspace{0.8cm}

% fig1
\item
Bare third order vertexes U and V:

\begin{picture}(24,30)(0,10)
\put(0,12){\line(1,0){12}}
\multiput(13,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(13,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(13,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(13,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(23.1,1){\vector(1,-1){1}}
\put(23.1,23){\vector(1,1){1}}
\put(11,12){\vector(1,0){1}}

\put(0,14){1}
\put(18,22){2}
\put(17,0){3}
\end{picture}

\eeb
\[V^{k_1}_{k_2 k_3} \]
\eep
% U

\begin{picture}(24,30)(0,10)
\put(0,12){\line(1,0){12}}
\multiput(13,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(13,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(13,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(13,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(23.1,1){\vector(1,-1){1}}
\put(23.1,23){\vector(1,1){1}}
\put(1,12){\vector(-1,0){1}}

\put(0,14){1}
\put(18,22){2}
\put(17,0){3}
\end{picture}

\eeb
\[U_{k_1 k_2 k_3}\]
\eep
\vspace{0.8cm}


%fig 2
\item
Bare Green function $G(k,\omega)$.

\begin{picture}(24,30)(0,10)
\multiput(13,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){3}{\oval(2,1.6)[b]}
\put(12,12){\vector(-1,0){12}}
\put(12,16){\makebox(0,0){$k$, $\omega$}}
\end{picture}

\eeb
\[G(k,\omega)=\frac{1}{\omega-\omega_k}\]
\eep


%fig 4
\end{enumerate}
%\end{document}
Note, that each vertex has just one straight line and others are wavy.

In order to calculate fourth order interaction matrix element
we have to add to bare fourth order vertex all possible combinations of
lower order vertexes (third order in this particular case) connected
with Green function
in such a way,
that the resulting diagrams have 2 incoming and 2 outgoing wave vectors and
having no internal loops.

It is easy to see, that the only way to fulfill these requirements is
to connect 2 third order vertexes by one Green function.
As the result we have 6 topologically different arrangements.
The arguments $k$ and $\omega$  of internal Green function
should be calculated from resonant conditions(\ref{res4}).
Since we are on the resonant manifold it does not matter
do we calculate arguments $k$ and $\omega$  of Green function
from left or from right vertex, because it they both
give the same result. This reflects the fact that two ratios
in each line in square brackets
of  (\ref{T4_hat}) are equal to each others. This removes extra
$1/2$.

%add1
\begin{picture}(36,30)(0,10)
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\put(18,12){\line(1,0){6}}
\put(0,24){\vector(1,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(35.1,1){\vector(1,-1){1}}


\put(23,12){\vector(1,0){1}}
\put(4,0){2}
\put(3,24){1}
\put(31,24){3}
\put(30,0){4}
\put(18,16){\makebox(0,0){1+2}}
\end{picture}

\eb
\[
-V^{k_1+k_2}_{k_1 k_2}V^{k_3+k_4}_{k_3 k_4}
\left[\frac{1}{\omega_{k_1+k_2}-\omega_{k_1}-\omega_{k_2}}\right]
\]
   \ep

%add2
\begin{picture}(36,30)(0,10)
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\put(18,12){\line(1,0){6}}
\put(0,24){\vector(1,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(35.1,1){\vector(1,-1){1}}


\put(12.7,12.7){\vector(-1,-1){1}}
\put(4,0){2}
\put(3,24){1}
\put(31,24){3}
\put(30,0){4}
\put(18,16){\makebox(0,0){-1-2}}
\end{picture}

\eb
\[
-U_{-k_1-k_2 k_1 k_2}U_{-k_3-k_4 k_3 k_4}
\left[\frac{1}{\omega_{k_1+k_2}+\omega_{k_1}+\omega_{k_2}}\right]
\]
   \ep

%add3
\begin{picture}(36,30)(0,10)
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\put(18,12){\line(1,0){6}}
\put(0,24){\vector(1,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
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\end{picture}

\eb
\[
-V^{k_1}_{k_3 k_1-k_3}V^{k_4}_{k_2 k_4-k_2}
\left[\frac{1}{\omega_{k_3}+\omega_{k_1-k_3}-\omega_{k_1}}\right]
\]
   \ep

%add4
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\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
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\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
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\put(18,16){\makebox(0,0){3-1}}
\end{picture}

\eb
\[
-V^{k_2}_{k_4 k_2-k_4}V^{k_3}_{k_1 k_3-k_1}
\left[\frac{1}{\omega_{k_4}+\omega_{k_2-k_4}-\omega_{k_2}}\right]
\]
   \ep

%add5
\begin{picture}(36,30)(0,10)
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\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\put(18,12){\line(1,0){6}}
\put(0,24){\vector(1,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(35.1,1){\vector(1,-1){1}}
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\put(18,16){\makebox(0,0){1-4}}
\end{picture}

\eb
\[
-V^{k_1}_{k_4 k_1-k_4}V^{k_3}_{k_2 k_3-k_2}
\left[\frac{1}{\omega_{k_4}+\omega_{k_1-k_4}-\omega_{k_1}}\right]
\]
   \ep

%add6
\begin{picture}(36,30)(0,10)
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\put(18,12){\line(1,0){6}}
\put(0,24){\vector(1,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(35.1,1){\vector(1,-1){1}}
\put(24.7,12.7){\vector(-1,-1){1}}
\put(0.7,0.7){\vector(-1,-1){1}}

\put(12.7,12.7){\vector(-1,-1){1}}
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\put(3,24){1}
\put(31,24){2}
\put(30,0){3}
\put(18,16){\makebox(0,0){4-1}}
\end{picture}

\eb
\[
-V^{k_2}_{k_3 k_2-k_3}V^{k_4}_{k_1 k_4-k_1}
\left[\frac{1}{\omega_{k_3}+\omega_{k_2-k_3}-\omega_{k_2}}\right]
\]
   \ep


\section{Appendix}

Following the same steps as in the Appendix B,
we can construct diagrammatic series for
five order matrix element $T^{k k_1 k_2}_{k_3 k_4}$.
We have to combine all third and fourth order vertexes
in such a way that we have 2 "incoming" arguments and 3 "out-coming" arguments.
The result of course will be the same if we would consider
3 "incoming" arguments and 2 "out-coming" arguments.
Considering all possible topologies consistent with definitions of
vertexes and Green functions and without internal loops,
we conclude, that there exist 60 diagrams constructed from three third order
verticesand  2 Green functions and 20 diagrams constructed from
one three order vertex, one four order vertex and one Green function.
We call these two groups  "3+3+3" and "4+3" correspondingly.

Below are the diagrams and analytical expressions for "3+3+3" and "4+3" terms.
Together with the bare fifth order vertex $Q^{k_1 k_2 k_3}_{p q}$
this sum gives the full fifth-order
interaction matrix element $T^{k_1 k_2 k_3}_{p q}$ or
$T_{k_1 k_2 k_3}^{p q}$.
We used these expressions as an input to Matematica, therefore
notation here is slightly different.
% *********************************************
% *********************************************
% *********************************************
% *********************************************
% *********************************************
% *********************************************
% *********************************************
{\small
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\eb
\[ {{V({\it k2} + {\it k3},{\it k2},{\it k3})\,
      V(p + q,{\it k1},{\it k2} + {\it k3})\,V(p + q,p,q)}\over
    {\left( \omega ({\it k2}) + \omega ({\it k3}) -
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( \omega (p) + \omega (q) - \omega (p + q) \right) }} \]

\ep

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\eb
\[ -{{U({\it k1},-p - q,{\it k2} + {\it k3})\,U(p,q,-p - q)\,
       V({\it k2} + {\it k3},{\it k2},{\it k3})}\over
     {\left( \omega ({\it k2}) + \omega ({\it k3}) -
         \omega ({\it k2} + {\it k3}) \right) \,
       \left( \omega (p) + \omega (q) + \omega (p + q) \right) }} \]

\ep


%picture3
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\eb
\[ -{{U({\it k2},{\it k3},-{\it k2} - {\it k3})\,
       V({\it k1},p + q,-{\it k2} - {\it k3})\,V(p + q,p,q)}\over
     {\left( \omega ({\it k2}) + \omega ({\it k3}) +
         \omega ({\it k2} + {\it k3}) \right) \,
       \left( \omega (p) + \omega (q) - \omega (p + q) \right) }} \]

\ep

%picture4
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\eb
 \[ {{U({\it k2},{\it k3},-{\it k2} - {\it k3})\,U(p,q,-p - q)\,
      V(-{\it k2} - {\it k3},{\it k1},-p - q)}\over
    {\left( \omega ({\it k2}) + \omega ({\it k3}) +
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( \omega (p) + \omega (q) + \omega (p + q) \right) }} \]

\ep

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\eb
\[ {{V({\it k2},p,{\it k2} - p)\,V(q,{\it k3},-{\it k3} + q)\,
      V(-{\it k3} + q,{\it k1},{\it k2} - p)}\over
    {\left( \omega ({\it k2}) - \omega ({\it k2} - p) - \omega (p) \right) \,\left( -\omega ({\it k3}) - \omega ({\it k3} - q) +
        \omega (q) \right) }} \]
\ep

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\eb
 \[ {{U({\it k1},{\it k3} - q,{\it k2} - p)\,V({\it k2},p,{\it k2} - p)\,
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\ep

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\eb
 \[ {{V({\it k1},-{\it k3} + q,-{\it k2} + p)\,
      V(p,{\it k2},-{\it k2} + p)\,V(q,{\it k3},-{\it k3} + q)}\over
    {\left( -\omega ({\it k2}) - \omega ({\it k2} - p) + \omega (p) \right) \,\left( -\omega ({\it k3}) - \omega ({\it k3} - q) +
        \omega (q) \right) }} \]
\ep

%picture8
\begin{picture}(48,24)(0,10)
%\input{pic2}
%\input{pic1-48}
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\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
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\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
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\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
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\put(17,16){\makebox(0,0){$3-q$}}
\put(31,16){\makebox(0,0){$p-2$}}
\end{picture}

\eb
 \[ {{V({\it k3},q,{\it k3} - q)\,V(p,{\it k2},-{\it k2} + p)\,
      V(-{\it k2} + p,{\it k1},{\it k3} - q)}\over
    {\left( -\omega ({\it k2}) - \omega ({\it k2} - p) + \omega (p) \right) \,\left( \omega ({\it k3}) - \omega ({\it k3} - q) -
        \omega (q) \right) }} \]
\ep

\newpage
%picture9
\begin{picture}(48,24)(0,10)
%\input{pic3}
%\input{pic1-48}
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\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
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\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
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\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
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\put(42,24){3}
\put(42,0){$p$}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
%\put(13,13){\vector(-1,-1){1}}
%\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-2$}}
\put(31,16){\makebox(0,0){$3-p$}}
\end{picture}

\eb
\[ {{V({\it k3},p,{\it k3} - p)\,V(q,{\it k2},-{\it k2} + q)\,
      V(-{\it k2} + q,{\it k1},{\it k3} - p)}\over
    {\left( \omega ({\it k3}) - \omega ({\it k3} - p) - \omega (p) \right) \,\left( -\omega ({\it k2}) - \omega ({\it k2} - q) +
        \omega (q) \right) }} \]
\ep

%picture10
\begin{picture}(48,24)(0,10)
%\input{pic3}
%\input{pic1-48}
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\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
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\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
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\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
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\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
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\put(42,24){3}
\put(42,0){$p$}
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\put(17,16){\makebox(0,0){$2-q$}}
\put(31,16){\makebox(0,0){$3-p$}}
\end{picture}

\eb
 \[ {{U({\it k1},{\it k2} - q,{\it k3} - p)\,V({\it k2},q,{\it k2} - q)\,
      V({\it k3},p,{\it k3} - p)}\over
    {\left( \omega ({\it k3}) - \omega ({\it k3} - p) - \omega (p) \right) \,\left( \omega ({\it k2}) - \omega ({\it k2} - q) -
        \omega (q) \right) }} \]
\ep

%picture11
\begin{picture}(48,24)(0,10)
%\input{pic3}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
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\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){2}
\put(25,24){1}
\put(42,24){3}
\put(42,0){$p$}
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\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-2$}}
\put(31,16){\makebox(0,0){$p-3$}}
\end{picture}
\eb
 \[ {{V({\it k1},-{\it k2} + q,-{\it k3} + p)\,
      V(p,{\it k3},-{\it k3} + p)\,V(q,{\it k2},-{\it k2} + q)}\over
    {\left( -\omega ({\it k3}) - \omega ({\it k3} - p) + \omega (p) \right) \,\left( -\omega ({\it k2}) - \omega ({\it k2} - q) +
        \omega (q) \right) }} \]
\ep

%picture12
\begin{picture}(48,24)(0,10)
%\input{pic3}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){2}
\put(25,24){1}
\put(42,24){3}
\put(42,0){$p$}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$2-q$}}
\put(31,16){\makebox(0,0){$p-3$}}
\end{picture}

\eb
 \[ {{V({\it k2},q,{\it k2} - q)\,V(p,{\it k3},-{\it k3} + p)\,
      V(-{\it k3} + p,{\it k1},{\it k2} - q)}\over
    {\left( -\omega ({\it k3}) - \omega ({\it k3} - p) + \omega (p) \right) \,\left( \omega ({\it k2}) - \omega ({\it k2} - q) -
        \omega (q) \right) }} \]
\ep

%picture13
\begin{picture}(48,24)(0,10)
%\input{pic4}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(10.85,13.15){\vector(1,-1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){$p$}
\put(25,24){2}
\put(42,24){1}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$p+q$}}
\put(31,16){\makebox(0,0){1+3}}
\end{picture}

\eb
\[ {{V({\it k1} + {\it k3},{\it k1},{\it k3})\,
      V(p + q,{\it k2},{\it k1} + {\it k3})\,V(p + q,p,q)}\over
    {\left( \omega ({\it k1}) + \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( \omega (p) + \omega (q) - \omega (p + q) \right) }} \]

\ep

%picture14
\begin{picture}(48,24)(0,10)
%\input{pic4}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(10.85,13.15){\vector(1,-1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){$p$}
\put(25,24){2}
\put(42,24){1}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(16,16){\makebox(0,0){$-p-q$}}
\put(31,16){\makebox(0,0){1+3}}
\end{picture}
\eb
 \[ {{U({\it k2},{\it k1} + {\it k3},-p - q)\,U(p,q,-p - q)\,
      V({\it k1} + {\it k3},{\it k1},{\it k3})}\over
    {\left( \omega ({\it k1}) + \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( -\omega (p) - \omega (q) - \omega (p + q) \right) }} \]

\ep

%picture15
\begin{picture}(48,24)(0,10)
%\input{pic4}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(10.85,13.15){\vector(1,-1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){$p$}
\put(25,24){2}
\put(42,24){1}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$p+q$}}
\put(32,16){\makebox(0,0){$-1-3$}}
\end{picture}
\eb
 \[ {{U({\it k1},{\it k3},-{\it k1} - {\it k3})\,
      V({\it k2},p + q,-{\it k1} - {\it k3})\,V(p + q,p,q)}\over
    {\left( -\omega ({\it k1}) - \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( \omega (p) + \omega (q) - \omega (p + q) \right) }} \]

\ep

%picture16
\begin{picture}(48,24)(0,10)
%\input{pic4}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(10.85,13.15){\vector(1,-1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){$p$}
\put(25,24){2}
\put(42,24){1}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(16,16){\makebox(0,0){$-p-q$}}
\put(32,16){\makebox(0,0){$-1-3$}}
\end{picture}

\eb
  \[ {{U(-{\it k1} - {\it k3},{\it k1},{\it k3})\,U(p,q,-p - q)\,
      V(-{\it k1} - {\it k3},{\it k2},-p - q)}\over
    {\left( -\omega ({\it k1}) - \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( -\omega (p) - \omega (q) - \omega (p + q) \right) }} \]

\ep

\newpage
%picture17
\begin{picture}(48,24)(0,10)
%\input{pic5}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
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\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
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\put(18,12){\line(1,0){6}}
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\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){1}
\put(25,24){2}
\put(42,24){3}
\put(42,0){$p$}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
%\put(13,13){\vector(-1,-1){1}}
%\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-1$}}
\put(31,16){\makebox(0,0){$3-p$}}
\end{picture}

\eb
\[ {{V({\it k3},p,{\it k3} - p)\,V(q,{\it k1},-{\it k1} + q)\,
      V(-{\it k1} + q,{\it k2},{\it k3} - p)}\over
    {\left( \omega ({\it k3}) - \omega ({\it k3} - p) - \omega (p) \right) \,\left( -\omega ({\it k1}) + \omega (q) -
        \omega (-{\it k1} + q) \right) }} \]

\ep

%picture18
\begin{picture}(48,24)(0,10)
%\input{pic5}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){1}
\put(25,24){2}
\put(42,24){3}
\put(42,0){$p$}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$1-q$}}
\put(31,16){\makebox(0,0){$3-p$}}
\end{picture}

\eb
 \[ {{U({\it k2},{\it k1} - q,{\it k3} - p)\,V({\it k1},q,{\it k1} - q)\,
      V({\it k3},p,{\it k3} - p)}\over
    {\left( \omega ({\it k3}) - \omega ({\it k3} - p) - \omega (p) \right) \,\left( \omega ({\it k1}) - \omega (q) -
        \omega (-{\it k1} + q) \right) }} \]

\ep

%picture19
\begin{picture}(48,24)(0,10)
%\input{pic5}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){1}
\put(25,24){2}
\put(42,24){3}
\put(42,0){$p$}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-1$}}
\put(31,16){\makebox(0,0){$p-3$}}
\end{picture}
\eb
 \[ {{V({\it k2},-{\it k1} + q,-{\it k3} + p)\,
      V(p,{\it k3},-{\it k3} + p)\,V(q,{\it k1},-{\it k1} + q)}\over
    {\left( -\omega ({\it k3}) - \omega ({\it k3} - p) + \omega (p) \right) \,\left( -\omega ({\it k1}) + \omega (q) -
        \omega (-{\it k1} + q) \right) }} \]

\ep

%picture20
\begin{picture}(48,24)(0,10)
%\input{pic5}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){1}
\put(25,24){2}
\put(42,24){3}
\put(42,0){$p$}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$1-q$}}
\put(31,16){\makebox(0,0){$p-3$}}
\end{picture}

\eb
 \[ {{V({\it k1},q,{\it k1} - q)\,V(p,{\it k3},-{\it k3} + p)\,
      V(-{\it k3} + p,{\it k2},{\it k1} - q)}\over
    {\left( -\omega ({\it k3}) - \omega ({\it k3} - p) + \omega (p) \right) \,\left( \omega ({\it k1}) - \omega (q) -
        \omega (-{\it k1} + q) \right) }} \]

\ep

%picture21
\begin{picture}(48,24)(0,10)
%\input{pic6}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){3}
\put(25,24){2}
\put(42,24){1}
\put(42,0){$p$}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
%\put(13,13){\vector(-1,-1){1}}
%\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-3$}}
\put(31,16){\makebox(0,0){$1-p$}}
\end{picture}
%6-1
\eb
 \[ {{V({\it k1},p,{\it k1} - p)\,V(q,{\it k3},-{\it k3} + q)\,
      V(-{\it k3} + q,{\it k2},{\it k1} - p)}\over
    {\left( \omega ({\it k1}) - \omega ({\it k1} - p) - \omega (p) \right) \,\left( -\omega ({\it k3}) - \omega ({\it k3} - q) +
        \omega (q) \right) }} \]

\ep

%picture22
\begin{picture}(48,24)(0,10)
%\input{pic6}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){3}
\put(25,24){2}
\put(42,24){1}
\put(42,0){$p$}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$3-q$}}
\put(31,16){\makebox(0,0){$1-p$}}
\end{picture}
\eb
\[ {{U({\it k2},{\it k3} - q,{\it k1} - p)\,V({\it k1},p,{\it k1} - p)\,
      V({\it k3},{\it k3} - q,q)}\over
    {\left( \omega ({\it k1}) - \omega ({\it k1} - p) - \omega (p) \right) \,\left( \omega ({\it k3}) - \omega ({\it k3} - q) -
        \omega (q) \right) }} \]

\ep

%picture23
\begin{picture}(48,24)(0,10)
%\input{pic6}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){3}
\put(25,24){2}
\put(42,24){1}
\put(42,0){$p$}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-3$}}
\put(31,16){\makebox(0,0){$p-1$}}
\end{picture}
\eb
\[ {{V({\it k2},-{\it k3} + q,-{\it k1} + p)\,
      V(p,{\it k1},-{\it k1} + p)\,V(q,{\it k3},-{\it k3} + q)}\over
    {\left( -\omega ({\it k1}) - \omega ({\it k1} - p) + \omega (p) \right) \,\left( -\omega ({\it k3}) - \omega ({\it k3} - q) +
        \omega (q) \right) }} \]

\ep

%picture24
\begin{picture}(48,24)(0,10)
%\input{pic6}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){3}
\put(25,24){2}
\put(42,24){1}
\put(42,0){$p$}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$3-q$}}
\put(31,16){\makebox(0,0){$p-1$}}
\end{picture}

\eb
 \[ {{V({\it k3},{\it k3} - q,q)\,V(p,{\it k1},-{\it k1} + p)\,
      V(-{\it k1} + p,{\it k3} - q,{\it k2})}\over
    {\left( -\omega ({\it k1}) - \omega ({\it k1} - p) + \omega (p) \right) \,\left( \omega ({\it k3}) - \omega ({\it k3} - q) -
        \omega (q) \right) }} \]

\ep

\newpage
%picture25
\begin{picture}(48,24)(0,10)
%\input{pic7}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(10.85,13.15){\vector(1,-1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){$p$}
\put(25,24){3}
\put(42,24){1}
\put(42,0){2}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$p+q$}}
\put(31,16){\makebox(0,0){1+2}}
\end{picture}
% 7 1
\eb
\[ {{V({\it k1} + {\it k2},{\it k1},{\it k2})\,
      V(p + q,{\it k3},{\it k1} + {\it k2})\,V(p + q,p,q)}\over
    {\left( \omega ({\it k1}) + \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( \omega (p) + \omega (q) - \omega (p + q) \right) }} \]

\ep

%picture26
\begin{picture}(48,24)(0,10)
%\input{pic7}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(10.85,13.15){\vector(1,-1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){$p$}
\put(25,24){3}
\put(42,24){1}
\put(42,0){2}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(16,16){\makebox(0,0){$-p-q$}}
\put(31,16){\makebox(0,0){1+2}}
\end{picture}
\eb
 \[ {{U({\it k3},{\it k1} + {\it k2},-p - q)\,U(p,q,-p - q)\,
      V({\it k1} + {\it k2},{\it k1},{\it k2})}\over
    {\left( \omega ({\it k1}) + \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( -\omega (p) - \omega (q) - \omega (p + q) \right) }} \]

\ep

%picture27
\begin{picture}(48,24)(0,10)
%\input{pic7}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(10.85,13.15){\vector(1,-1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){$p$}
\put(25,24){3}
\put(42,24){1}
\put(42,0){2}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$p+q$}}
\put(32,16){\makebox(0,0){$-1-2$}}
\end{picture}
\eb
\[ {{U({\it k1},{\it k2},-{\it k1} - {\it k2})\,
      V({\it k3},p + q,-{\it k1} - {\it k2})\,V(p + q,p,q)}\over
    {\left( -\omega ({\it k1}) - \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( \omega (p) + \omega (q) - \omega (p + q) \right) }} \]

\ep
%picture28
\begin{picture}(48,24)(0,10)
%\input{pic7}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(10.85,13.15){\vector(1,-1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){$p$}
\put(25,24){3}
\put(42,24){1}
\put(42,0){2}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(16,16){\makebox(0,0){$-p-q$}}
\put(32,16){\makebox(0,0){$-1-2$}}
\end{picture}

\eb
\[ {{U({\it k1},{\it k2},-{\it k1} - {\it k2})\,U(p,q,-p - q)\,
      V(-{\it k1} - {\it k2},{\it k3},-p - q)}\over
    {\left( -\omega ({\it k1}) - \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( -\omega (p) - \omega (q) - \omega (p + q) \right) }} \]

\ep

%picture29
\begin{picture}(48,24)(0,10)
%\input{pic8}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
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\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){1}
\put(25,24){3}
\put(42,24){2}
\put(42,0){$p$}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
%\put(13,13){\vector(-1,-1){1}}
%\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-1$}}
\put(31,16){\makebox(0,0){$2-p$}}
\end{picture}

\eb
\[ {{V({\it k2},p,{\it k2} - p)\,V(q,{\it k1},-{\it k1} + q)\,
      V(-{\it k1} + q,{\it k3},{\it k2} - p)}\over
    {\left( \omega ({\it k2}) - \omega ({\it k2} - p) - \omega (p) \right) \,\left( -\omega ({\it k1}) + \omega (q) -
        \omega (-{\it k1} + q) \right) }} \]

\ep

%picture30
\begin{picture}(48,24)(0,10)
%\input{pic8}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){1}
\put(25,24){3}
\put(42,24){2}
\put(42,0){$p$}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$1-q$}}
\put(31,16){\makebox(0,0){$2-p$}}
\end{picture}
\eb
\[ {{U({\it k1} - q,{\it k3},{\it k2} - p)\,V({\it k1},q,{\it k1} - q)\,
      V({\it k2},p,{\it k2} - p)}\over
    {\left( \omega ({\it k2}) - \omega ({\it k2} - p) - \omega (p) \right) \,\left( \omega ({\it k1}) - \omega (q) -
        \omega (-{\it k1} + q) \right) }} \]

\ep

%picture31
\begin{picture}(48,24)(0,10)
%\input{pic8}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){1}
\put(25,24){3}
\put(42,24){2}
\put(42,0){$p$}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-1$}}
\put(31,16){\makebox(0,0){$p-2$}}
\end{picture}

\eb
 \[ {{V({\it k3},-{\it k1} + q,-{\it k2} + p)\,
      V(p,{\it k2},-{\it k2} + p)\,V(q,{\it k1},-{\it k1} + q)}\over
    {\left( -\omega ({\it k2}) - \omega ({\it k2} - p) + \omega (p) \right) \,\left( -\omega ({\it k1}) + \omega (q) -
        \omega (-{\it k1} + q) \right) }} \]

\ep

%picture32
\begin{picture}(48,24)(0,10)
%\input{pic8}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){1}
\put(25,24){3}
\put(42,24){2}
\put(42,0){$p$}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$1-q$}}
\put(31,16){\makebox(0,0){$p-2$}}
\end{picture}

\eb
 \[ {{V({\it k1},q,{\it k1} - q)\,V(p,{\it k2},-{\it k2} + p)\,
      V(-{\it k2} + p,{\it k3},{\it k1} - q)}\over
    {\left( -\omega ({\it k2}) - \omega ({\it k2} - p) + \omega (p) \right) \,\left( \omega ({\it k1}) - \omega (q) -
        \omega (-{\it k1} + q) \right) }} \]

\ep

\newpage
%picture33
\begin{picture}(48,24)(0,10)
%\input{pic9}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
\put(3,24){2}
\put(25,24){3}
\put(42,24){1}
\put(42,0){$p$}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
%\put(13,13){\vector(-1,-1){1}}
%\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-2$}}
\put(31,16){\makebox(0,0){$1-p$}}
\end{picture}

\eb
 \[ {{V({\it k1},p,{\it k1} - p)\,V(q,{\it k2},-{\it k2} + q)\,
      V(-{\it k2} + q,{\it k3},{\it k1} - p)}\over
    {\left( \omega ({\it k1}) - \omega ({\it k1} - p) - \omega (p) \right) \,\left( -\omega ({\it k2}) - \omega ({\it k2} - q) +
        \omega (q) \right) }} \]

\ep

%picture34
\begin{picture}(48,24)(0,10)
%\input{pic9}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
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\put(25,24){3}
\put(42,24){1}
\put(42,0){$p$}
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\put(17,16){\makebox(0,0){$2-q$}}
\put(31,16){\makebox(0,0){$1-p$}}
\end{picture}
\eb
\[ {{U({\it k3},{\it k2} - q,{\it k1} - p)\,V({\it k1},p,{\it k1} - p)\,
      V({\it k2},q,{\it k2} - q)}\over
    {\left( \omega ({\it k1}) - \omega ({\it k1} - p) - \omega (p) \right) \,\left( \omega ({\it k2}) - \omega ({\it k2} - q) -
        \omega (q) \right) }} \]

\ep

%picture35
\begin{picture}(48,24)(0,10)
%\input{pic9}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
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\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(36.9,10.85){\vector(-1,1){1}}
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\put(42,24){1}
\put(42,0){$p$}
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\put(17,16){\makebox(0,0){$q-2$}}
\put(31,16){\makebox(0,0){$p-1$}}
\end{picture}
\eb
\[ {{V({\it k3},-{\it k2} + q,-{\it k1} + p)\,
      V(p,{\it k1},-{\it k1} + p)\,V(q,{\it k2},-{\it k2} + q)}\over
    {\left( -\omega ({\it k1}) - \omega ({\it k1} - p) + \omega (p) \right) \,\left( -\omega ({\it k2}) - \omega ({\it k2} - q) +
        \omega (q) \right) }} \]

\ep

%picture36
\begin{picture}(48,24)(0,10)
%\input{pic9}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
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\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
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\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
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%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
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\put(42,24){1}
\put(42,0){$p$}
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\put(17,16){\makebox(0,0){$2-q$}}
\put(31,16){\makebox(0,0){$p-1$}}
\end{picture}

\eb
 \[ {{V({\it k2},q,{\it k2} - q)\,V(p,{\it k1},-{\it k1} + p)\,
      V(-{\it k1} + p,{\it k3},{\it k2} - q)}\over
    {\left( -\omega ({\it k1}) - \omega ({\it k1} - p) + \omega (p) \right) \,\left( \omega ({\it k2}) - \omega ({\it k2} - q) -
        \omega (q) \right) }} \]

\ep

%picture37
\begin{picture}(48,24)(0,10)
%\input{pic10}
%\input{pic1-48}
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\put(0,0){\line(1,1){12}}
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\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){1}
\put(25,24){$p$}
\put(42,24){2}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$q-1$}}
\put(31,16){\makebox(0,0){2+3}}
\end{picture}
\eb
 \[ {{V({\it k2} + {\it k3},{\it k2},{\it k3})\,
      V({\it k2} + {\it k3},p,-{\it k1} + q)\,V(q,{\it k1},-{\it k1} + q)}\over
 {\left( \omega ({\it k2}) + \omega ({\it k3}) -
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( -\omega ({\it k1}) + \omega (q) - \omega (-{\it k1} + q) \right) }} \]
  \ep

%picture38
\begin{picture}(48,24)(0,10)
%\input{pic10}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){1}
\put(25,24){$p$}
\put(42,24){2}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$1-q$}}
\put(31,16){\makebox(0,0){2+3}}
\end{picture}
\eb
\[ {{V({\it k1},q,{\it k1} - q)\,
      V({\it k2} + {\it k3},{\it k2},{\it k3})\,
      V(p,{\it k1} - q,{\it k2} + {\it k3})}\over
    {\left( \omega ({\it k2}) + \omega ({\it k3}) -
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( \omega ({\it k1}) - \omega (q) - \omega (-{\it k1} + q) \right) }} \]
   \ep

%picture39
\begin{picture}(48,24)(0,10)
%\input{pic10}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){1}
\put(25,24){$p$}
\put(42,24){2}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-1$}}
\put(32,16){\makebox(0,0){$-2-3$}}
\end{picture}
\eb
 \[ {{U({\it k2},{\it k3},-{\it k2} - {\it k3})\,
      U(p,-{\it k1} + q,-{\it k2} - {\it k3})\,V(q,{\it k1},-{\it k1} + q)}\
     \over {\left( -\omega ({\it k2}) - \omega ({\it k3}) -
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( -\omega ({\it k1}) + \omega (q) - \omega (-{\it k1} + q) \right) }} \]
 \ep

%picture40
\begin{picture}(48,24)(0,10)
%\input{pic10}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){1}
\put(25,24){$p$}
\put(42,24){2}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$1-q$}}
\put(32,16){\makebox(0,0){$-2-3$}}
\end{picture}

\eb
 \[ {{U({\it k2},{\it k3},-{\it k2} - {\it k3})\,
      V({\it k1},q,{\it k1} - q)\,V({\it k1} - q,p,-{\it k2} - {\it k3})}\over {\left( -\omega ({\it k2}) - \omega ({\it k3}) -
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( \omega ({\it k1}) - \omega (q) - \omega (-{\it k1} + q) \right) }} \]
  \ep

\newpage
%picture41
\begin{picture}(48,24)(0,10)
%\input{pic11}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){2}
\put(25,24){$p$}
\put(42,24){1}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$q-2$}}
\put(31,16){\makebox(0,0){1+3}}
\end{picture}
\eb
 \[ {{V({\it k1} + {\it k3},{\it k1},{\it k3})\,
      V({\it k1} + {\it k3},p,-{\it k2} + q)\,V(q,{\it k2},-{\it k2} + q)}
     \over {\left( \omega ({\it k1}) + \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( -\omega ({\it k2}) - \omega ({\it k2} - q) + \omega (q) \right) }} \]
  \ep

%picture42
\begin{picture}(48,24)(0,10)
%\input{pic11}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){2}
\put(25,24){$p$}
\put(42,24){1}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$2-q$}}
\put(31,16){\makebox(0,0){1+3}}
\end{picture}
\eb
 \[ {{V({\it k2},q,{\it k2} - q)\,
      V({\it k1} + {\it k3},{\it k1},{\it k3})\,
      V(p,{\it k2} - q,{\it k1} + {\it k3})}\over
    {\left( \omega ({\it k1}) + \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( \omega ({\it k2}) - \omega ({\it k2} - q) - \omega (q) \right) }} \]
  \ep

%picture43
\begin{picture}(48,24)(0,10)
%\input{pic11}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){2}
\put(25,24){$p$}
\put(42,24){1}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-2$}}
\put(32,16){\makebox(0,0){$-1-3$}}
\end{picture}
\eb
 \[ {{U({\it k1},{\it k3},-{\it k1} - {\it k3})\,
      U(p,-{\it k2} + q,-{\it k1} - {\it k3})\,V(q,{\it k2},-{\it k2} + q)}\
     \over {\left( -\omega ({\it k1}) - \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( -\omega ({\it k2}) - \omega ({\it k2} - q) + \omega (q) \right) }} \]
  \ep

%picture44
\begin{picture}(48,24)(0,10)
%\input{pic11}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){2}
\put(25,24){$p$}
\put(42,24){1}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$2-q$}}
\put(32,16){\makebox(0,0){$-1-3$}}
\end{picture}

\eb
 \[ -{{U({\it k1},{\it k3},-{\it k1} - {\it k3})\,
       V({\it k2},q,{\it k2} - q)\,V({\it k2} - q,p,-{\it k1} - {\it k3})}
      \over {\left( \omega ({\it k1}) + \omega ({\it k3}) +
         \omega ({\it k1} + {\it k3}) \right) \,
       \left( \omega ({\it k2}) - \omega ({\it k2} - q) - \omega (q) \right) }} \]
  \ep

%picture45
\begin{picture}(48,24)(0,10)
%\input{pic12}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){3}
\put(25,24){$p$}
\put(42,24){1}
\put(42,0){2}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$q-3$}}
\put(31,16){\makebox(0,0){1+2}}
\end{picture}

\eb
 \[ {{V({\it k1} + {\it k2},{\it k1},{\it k2})\,
      V({\it k1} + {\it k2},p,-{\it k3} + q)\,V(q,{\it k3},-{\it k3} + q)}
    \over {\left( \omega ({\it k1}) + \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( -\omega ({\it k3}) - \omega ({\it k3} - q) + \omega (q) \right) }} \]
   \ep

%picture46
\begin{picture}(48,24)(0,10)
%\input{pic12}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){3}
\put(25,24){$p$}
\put(42,24){1}
\put(42,0){2}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$3-q$}}
\put(31,16){\makebox(0,0){1+2}}
\end{picture}
\eb
 \[ {{V({\it k1} + {\it k2},{\it k1},{\it k2})\,
      V({\it k3},q,{\it k3} - q)\,V(p,{\it k3} - q,{\it k1} + {\it k2})}\over {\left( \omega ({\it k1}) + \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( \omega ({\it k3}) - \omega ({\it k3} - q) - \omega (q) \right) }} \]
 \ep

%picture47
\begin{picture}(48,24)(0,10)
%\input{pic12}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){3}
\put(25,24){$p$}
\put(42,24){1}
\put(42,0){2}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$q-3$}}
\put(32,16){\makebox(0,0){$-1-2$}}
\end{picture}
\eb
\[ {{U(-{\it k1} - {\it k2},{\it k1},{\it k2})\,
      U(-{\it k3} + q,-{\it k1} - {\it k2},p)\,V(q,{\it k3},-{\it k3} + q)}\
     \over {\left( -\omega ({\it k1}) - \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( -\omega ({\it k3}) - \omega ({\it k3} - q) + \omega (q) \right) }} \]
  \ep

%picture48
\begin{picture}(48,24)(0,10)
%\input{pic12}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(23.1,13.){\vector(1,-1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){3}
\put(25,24){$p$}
\put(42,24){1}
\put(42,0){2}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$3-q$}}
\put(32,16){\makebox(0,0){$-1-2$}}
\end{picture}

\eb
 \[ -{{U({\it k1},{\it k2},-{\it k1} - {\it k2})\,
       V({\it k3},q,{\it k3} - q)\,V({\it k3} - q,p,-{\it k1} - {\it k2})}
      \over {\left( \omega ({\it k1}) + \omega ({\it k2}) +
         \omega ({\it k1} + {\it k2}) \right) \,
       \left( \omega ({\it k3}) - \omega ({\it k3} - q) - \omega (q) \right) }} \]
  \ep

\newpage
%picture49
\begin{picture}(48,24)(0,10)
%\input{pic13}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){1}
\put(42,24){2}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$p-1$}}
\put(31,16){\makebox(0,0){2+3}}
\end{picture}

\eb
 \[ {{V({\it k2} + {\it k3},{\it k2},{\it k3})\,
      V({\it k2} + {\it k3},q,-{\it k1} + p)\,V(p,{\it k1},-{\it k1} + p)}
     \over {\left( \omega ({\it k2}) + \omega ({\it k3}) -
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( -\omega ({\it k1}) - \omega ({\it k1} - p) + \omega (p) \right) }} \]
  \ep

%picture50
\begin{picture}(48,24)(0,10)
%\input{pic13}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){1}
\put(42,24){2}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$1-p$}}
\put(31,16){\makebox(0,0){2+3}}
\end{picture}
\eb
 \[ {{V({\it k1},p,{\it k1} - p)\,
      V({\it k2} + {\it k3},{\it k2},{\it k3})\,
      V(q,{\it k1} - p,{\it k2} + {\it k3})}\over
    {\left( \omega ({\it k2}) + \omega ({\it k3}) -
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( \omega ({\it k1}) - \omega ({\it k1} - p) - \omega (p) \right) }} \]
  \ep

%picture51
\begin{picture}(48,24)(0,10)
%\input{pic13}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){1}
\put(42,24){2}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$p-1$}}
\put(32,16){\makebox(0,0){$-2-3$}}
\end{picture}
\eb
\[ {{U({\it k2},{\it k3},-{\it k2} - {\it k3})\,
      U(q,-{\it k1} + p,-{\it k2} - {\it k3})\,V(p,{\it k1},-{\it k1} + p)}\
     \over {\left( -\omega ({\it k2}) - \omega ({\it k3}) -
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( -\omega ({\it k1}) - \omega ({\it k1} - p) + \omega (p) \right) }} \]
  \ep

%picture52
\begin{picture}(48,24)(0,10)
%\input{pic13}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){1}
\put(42,24){2}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$1-p$}}
\put(32,16){\makebox(0,0){$-2-3$}}
\end{picture}

\eb
 \[ {{U(-{\it k2} - {\it k3},{\it k2},{\it k3})\,
      V({\it k1},p,{\it k1} - p)\,V({\it k1} - p,q,-{\it k2} - {\it k3})}\over {\left( -\omega ({\it k2}) - \omega ({\it k3}) -
        \omega ({\it k2} + {\it k3}) \right) \,
      \left( \omega ({\it k1}) - \omega ({\it k1} - p) - \omega (p) \right) }} \]
 \ep

%picture53
\begin{picture}(48,24)(0,10)
%\input{pic14}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){2}
\put(42,24){1}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$p-2$}}
\put(31,16){\makebox(0,0){1+3}}
\end{picture}

\eb
 \[ {{V({\it k1} + {\it k3},{\it k1},{\it k3})\,
      V({\it k1} + {\it k3},q,-{\it k2} + p)\,V(p,{\it k2},-{\it k2} + p)}\
     \over {\left( \omega ({\it k1}) + \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( -\omega ({\it k2}) - \omega ({\it k2} - p) + \omega (p) \right) }} \]
   \ep

%picture54
\begin{picture}(48,24)(0,10)
%\input{pic14}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){2}
\put(42,24){1}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$2-p$}}
\put(31,16){\makebox(0,0){1+3}}
\end{picture}
\eb
\[ {{V({\it k2},p,{\it k2} - p)\,
      V({\it k1} + {\it k3},{\it k1},{\it k3})\,
      V(q,{\it k2} - p,{\it k1} + {\it k3})}\over
    {\left( \omega ({\it k1}) + \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( \omega ({\it k2}) - \omega ({\it k2} - p) - \omega (p) \right) }} \]
  \ep

%picture55
\begin{picture}(48,24)(0,10)
%\input{pic14}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){2}
\put(42,24){1}
\put(42,0){3}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$p-2$}}
\put(32,16){\makebox(0,0){$-1-3$}}
\end{picture}
\eb
 \[ {{U(-{\it k1} - {\it k3},{\it k1},{\it k3})\,
      U(-{\it k2} + p,q,-{\it k1} - {\it k3})\,V(p,{\it k2},-{\it k2} + p)}\
     \over {\left( -\omega ({\it k1}) - \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( -\omega ({\it k2}) - \omega ({\it k2} - p) + \omega (p) \right) }} \]
  \ep

%picture56
\begin{picture}(48,24)(0,10)
%\input{pic14}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){2}
\put(42,24){1}
\put(42,0){3}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$2-p$}}
\put(32,16){\makebox(0,0){$-1-3$}}
\end{picture}

\eb
 \[ {{U({\it k1},{\it k3},-{\it k1} - {\it k3})\,
      V({\it k2},p,{\it k2} - p)\,V({\it k2} - p,q,-{\it k1} - {\it k3})}\over {\left( -\omega ({\it k1}) - \omega ({\it k3}) -
        \omega ({\it k1} + {\it k3}) \right) \,
      \left( \omega ({\it k2}) - \omega ({\it k2} - p) - \omega (p) \right) }} \]
   \ep

\newpage
%picture57
\begin{picture}(48,24)(0,10)
%\input{pic15}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){3}
\put(42,24){1}
\put(42,0){2}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$p-3$}}
\put(31,16){\makebox(0,0){1+2}}
\end{picture}

\eb
 \[ {{V({\it k1} + {\it k2},{\it k1},{\it k2})\,
      V({\it k1} + {\it k2},q,-{\it k3} + p)\,V(p,{\it k3},-{\it k3} + p)}\
     \over {\left( \omega ({\it k1}) + \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( -\omega ({\it k3}) - \omega ({\it k3} - p) + \omega (p) \right) }} \]
   \ep

%picture58
\begin{picture}(48,24)(0,10)
%\input{pic15}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){3}
\put(42,24){1}
\put(42,0){2}
\put(13,13){\vector(-1,-1){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$3-p$}}
\put(31,16){\makebox(0,0){1+2}}
\end{picture}

% this was corrected December 5 94
\eb
 \[ {{V({\it k3},{p},{\it k3}-p})\,
      V(q, {\it k3} -p, {\it k1}+{\it k2})\,
V({\it k1}+{\it k2},{\it k1},{\it k2})}\
     \over {\left( \omega ({\it k1}) + \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( \omega ({\it k3}) - \omega ({\it k3} - p) - \omega (p) \right) } \]
\ep

%picture59
\begin{picture}(48,24)(0,10)
%\input{pic15}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){3}
\put(42,24){1}
\put(42,0){2}
\put(22.5,12){\vector(1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$p-3$}}
\put(32,16){\makebox(0,0){$-1-2$}}
\end{picture}
\eb
\[ {{U({\it k1},{\it k2},-{\it k1} - {\it k2})\,
      U(q,-{\it k3} + p,-{\it k1} - {\it k2})\,V(p,{\it k3},-{\it k3} + p)}\
     \over {\left( -\omega ({\it k1}) - \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( -\omega ({\it k3}) - \omega ({\it k3} - p) + \omega (p) \right) }} \]
   \ep

%picture60
\begin{picture}(48,24)(0,10)
%\input{pic15}
%\input{pic49-60}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\put(24,24){\vector(0,-1){12}}
\multiput(11,12)(-2,-2){6}{\oval(2,2)[b,r]}
\multiput(11,10)(-2,-2){6}{\oval(2,2)[t,l]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(10.85,10.85){\vector(1,1){1}}
\put(.9,23){\vector(-1,1){1}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,0){$p$}
\put(25,24){$q$}
\put(3,24){3}
\put(42,24){1}
\put(42,0){2}
\put(13,13){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(17,16){\makebox(0,0){$3-p$}}
\put(32,16){\makebox(0,0){$-1-2$}}
\end{picture}
\eb
%This was corrected December 5 94X
\[ {V({\it k3},p,{\it k3}-p)\,
    V({\it k3}-p,q,-{\it k1}-{\it k2})\,U({\it k1},{\it k2},-{\it k1}-{\it k2)}\
     \over {\left( -\omega ({\it k1}) - \omega ({\it k2}) -
        \omega ({\it k1} + {\it k2}) \right) \,
      \left( \omega ({\it k3}) - \omega ({\it k3} - p) - \omega (p) \right) }} \]
   \ep

%picture61
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(11,12.8){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(35.1,1){\vector(1,-1){1}}
\put(23,12){\vector(1,0){1}}
\put(1,8){$p$}
\put(13,24){1}
\put(30,24){2}
\put(30,0){3}
\put(19,16){\makebox(0,0){$2+3$}}
\end{picture}

\eb
\[ {{V({\it k2} + {\it k3},{\it k2},{\it k3})\,
      W(p,q,{\it k1},{\it k2} + {\it k3})}\over
    {\omega ({\it k2}) + \omega ({\it k3}) - \omega ({\it k2} + {\it k3})}}
   \]
   \ep

%picture62
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(11,12.8){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(35.1,1){\vector(1,-1){1}}
\put(13.2,11){\vector(-1,1){1}}
\put(1,8){$p$}
\put(13,24){1}
\put(30,24){3}
\put(30,0){2}
\put(20,16){\makebox(0,0){$-2-3$}}
\end{picture}

\eb
 \[ -{{G({\it k1},p,q,-{\it k2} - {\it k3})\,
       U({\it k2},{\it k3},-{\it k2} - {\it k3})}\over
     {\omega ({\it k2}) + \omega ({\it k3}) + \omega ({\it k2} + {\it k3})}}
   \]
  \ep

%picture63
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(11,12.8){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(35.1,1){\vector(1,-1){1}}
\put(23,12){\vector(1,0){1}}
\put(1,8){$p$}
\put(13,24){2}
\put(30,24){3}
\put(30,0){1}
\put(19,16){\makebox(0,0){$1+3$}}
\end{picture}

\eb
 \[ {{V({\it k1} + {\it k3},{\it k1},{\it k3})\,
      W(p,q,{\it k2},{\it k1} + {\it k3})}\over
    {\omega ({\it k1}) + \omega ({\it k3}) - \omega ({\it k1} + {\it k3})}}
   \]
  \ep

%picture64
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(11,12.8){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(35.1,1){\vector(1,-1){1}}
\put(13.2,11){\vector(-1,1){1}}
\put(1,8){$p$}
\put(13,24){2}
\put(30,24){3}
\put(30,0){1}
\put(20,16){\makebox(0,0){$-1-3$}}
\end{picture}

\eb
\[ -{{G({\it k2},p,q,-{\it k1} - {\it k3})\,
       U({\it k1},{\it k3},-{\it k1} - {\it k3})}\over
     {\omega ({\it k1}) + \omega ({\it k3}) + \omega ({\it k1} + {\it k3})}}
   \]
   \ep

\newpage
%picture65
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(11,12.8){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(35.1,1){\vector(1,-1){1}}
\put(23,12){\vector(1,0){1}}
\put(1,8){$p$}
\put(13,24){3}
\put(30,24){2}
\put(30,0){1}
\put(19,16){\makebox(0,0){$1+2$}}
\end{picture}

\eb
 \[ {{V({\it k1} + {\it k2},{\it k1},{\it k2})\,
      W(p,q,{\it k3},{\it k1} + {\it k2})}\over
    {\omega ({\it k1}) + \omega ({\it k2}) - \omega ({\it k1} + {\it k2})}}
   \]
   \ep

%picture66
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(11,12.8){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(35.1,1){\vector(1,-1){1}}
\put(13.2,11){\vector(-1,1){1}}
\put(1,8){$p$}
\put(13,24){3}
\put(30,24){2}
\put(30,0){1}
\put(20,16){\makebox(0,0){$-1-2$}}
\end{picture}

\eb
 \[ -{{G({\it k3},p,q,-{\it k1} - {\it k2})\,
       U(-{\it k1} - {\it k2},{\it k1},{\it k2})}\over
     {\omega ({\it k1}) + \omega ({\it k2}) + \omega ({\it k1} + {\it k2})}}
   \]
   \ep

%picture67
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(2.2,12.8){\vector(-1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(25.2,11){\vector(-1,1){1}}
\put(23,12){\vector(1,0){1}}
\put(1,8){2}
\put(13,24){3}
\put(30,24){1}
\put(30,0){$p$}
\put(19,16){\makebox(0,0){$1-p$}}
\end{picture}

\eb
 \[ {{G(q,{\it k1} - p,{\it k2},{\it k3})\,V({\it k1},p,{\it k1} - p)}\over
    {\omega ({\it k1}) - \omega ({\it k1} - p) - \omega (p)}} \]

   \ep

%picture68
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(2.2,12.8){\vector(-1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(25.2,11){\vector(-1,1){1}}
\put(13.2,11){\vector(-1,1){1}}
\put(1,8){2}
\put(13,24){3}
\put(30,24){1}
\put(30,0){$p$}
\put(19,16){\makebox(0,0){$p-1$}}
\end{picture}

\eb
 \[ {{V(p,{\it k1},-{\it k1} + p)\,
      W(q,-{\it k1} + p,{\it k2},{\it k3})}\over
    {-\omega ({\it k1}) - \omega ({\it k1} - p) + \omega (p)}} \]

   \ep

%picture69
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(2.2,12.8){\vector(-1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(25.2,11){\vector(-1,1){1}}
\put(23,12){\vector(1,0){1}}
\put(1,8){1}
\put(13,24){3}
\put(30,24){2}
\put(30,0){$p$}
\put(19,16){\makebox(0,0){$2-p$}}
\end{picture}

\eb
 \[ {{G(q,{\it k2} - p,{\it k1},{\it k3})\,V({\it k2},p,{\it k2} - p)}\over
    {\omega ({\it k2}) - \omega ({\it k2} - p) - \omega (p)}} \]

   \ep

%picture70
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(2.2,12.8){\vector(-1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(25.2,11){\vector(-1,1){1}}
\put(13.2,11){\vector(-1,1){1}}
\put(1,8){1}
\put(13,24){3}
\put(30,24){2}
\put(30,0){$p$}
\put(19,16){\makebox(0,0){$p-2$}}
\end{picture}

\eb
 \[ {{V(p,{\it k2},-{\it k2} + p)\,
      W(q,-{\it k2} + p,{\it k1},{\it k3})}\over
    {-\omega ({\it k2}) - \omega ({\it k2} - p) + \omega (p)}} \]

   \ep

%picture71
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(2.2,12.8){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
%\put(25.2,11){\vector(-1,1){1}}
\put(23,12){\vector(1,0){1}}
\put(35.1,1){\vector(1,-1){1}}
\put(1,8){1}
\put(13,24){2}
\put(30,24){$p$}
\put(30,0){3}
\put(19,16){\makebox(0,0){$3-p$}}
\end{picture}

\eb
 \[ {{G(q,{\it k3} - p,{\it k1},{\it k2})\,V({\it k3},p,{\it k3} - p)}\over
    {\omega ({\it k3}) - \omega ({\it k3} - p) - \omega (p)}} \]

 \ep

%picture72
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(2.2,12.8){\vector(-1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
%\put(25.2,11){\vector(-1,1){1}}
\put(13.2,11){\vector(-1,1){1}}
\put(35.1,1){\vector(1,-1){1}}
\put(1,8){1}
\put(13,24){2}
\put(30,24){$p$}
\put(30,0){3}
\put(19,16){\makebox(0,0){$p-3$}}
\end{picture}

\eb
 \[ {{V(p,{\it k3},-{\it k3} + p)\,
      W(q,-{\it k3} + p,{\it k2},{\it k1})}\over
    {-\omega ({\it k3}) - \omega ({\it k3} - p) + \omega (p)}} \]

\ep

\newpage
%picture73
%\input{pic73-80}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(12,11)(0,-4){3}{\oval(1.6,2)[l]}
\multiput(12,9)(0,-4){2}{\oval(1.6,2)[r]}

\put(12,12){\line(1,0){6}}
\multiput(19,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(21,12)(4,0){1}{\oval(2,1.6)[b]}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\put(36,0){\vector(-1,1){12}}
\put(12,22){\vector(-1,1){1}}
\put(2.2,12.8){\vector(-1,-1){1}}
\put(32,0){$q$}


\put(12,2){\vector(1,-1){1}}
\put(23,13){\vector(1,-1){1}}
\put(25,13){\vector(-1,-1){1}}
\put(15,0){3}
\put(1,8){2}
\put(13,24){1}
\put(30,24){$p$}
\put(19,16){\makebox(0,0){$-q-p$}}
\end{picture}

\eb
 \[ -{{R({\it k1},{\it k2},{\it k3},-p - q)\,U(-p - q,p,q)}\over
     {\omega (p) + \omega (q) + \omega (p + q)}} \]

   \ep

%picture74
%\input{pic73-80}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(12,11)(0,-4){3}{\oval(1.6,2)[l]}
\multiput(12,9)(0,-4){2}{\oval(1.6,2)[r]}

\put(12,12){\line(1,0){6}}
\multiput(19,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(21,12)(4,0){1}{\oval(2,1.6)[b]}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\put(36,0){\vector(-1,1){12}}
\put(12,22){\vector(-1,1){1}}
\put(2.2,12.8){\vector(-1,-1){1}}
\put(32,0){$q$}


\put(12,2){\vector(1,-1){1}}
\put(13,12){\vector(-1,0){1}}
\put(25,13){\vector(-1,-1){1}}
\put(15,0){3}
\put(1,8){2}
\put(13,24){1}
\put(30,24){$p$}
\put(19,16){\makebox(0,0){$p+q$}}
\end{picture}

\eb
 \[ {{G(p + q,{\it k1},{\it k2},{\it k3})\,V(p + q,p,q)}\over
    {\omega (p) + \omega (q) - \omega (p + q)}} \]

  \ep

%picture75
%\input{pic73-80}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(12,11)(0,-4){3}{\oval(1.6,2)[l]}
\multiput(12,9)(0,-4){2}{\oval(1.6,2)[r]}

\put(12,12){\line(1,0){6}}
\multiput(19,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(21,12)(4,0){1}{\oval(2,1.6)[b]}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\put(36,0){\vector(-1,1){12}}
\put(12,22){\vector(-1,1){1}}
\put(2.2,12.8){\vector(-1,-1){1}}
\put(32,0){$q$}


\put(11,11){\vector(1,1){1}}
\put(23,13){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(15,0){$p$}
\put(1,8){1}
\put(13,24){2}
\put(30,24){3}
\put(19,16){\makebox(0,0){$3-q$}}
\end{picture}

\eb
\[ {{G(p,{\it k1},{\it k2},{\it k3} - q)\,V({\it k3},q,{\it k3} - q)}\over
    {\omega ({\it k3}) - \omega ({\it k3} - q) - \omega (q)}} \]

  \ep

%picture76
%\input{pic73-80}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(12,11)(0,-4){3}{\oval(1.6,2)[l]}
\multiput(12,9)(0,-4){2}{\oval(1.6,2)[r]}

\put(12,12){\line(1,0){6}}
\multiput(19,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(21,12)(4,0){1}{\oval(2,1.6)[b]}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\put(36,0){\vector(-1,1){12}}
\put(12,22){\vector(-1,1){1}}
\put(2.2,12.8){\vector(-1,-1){1}}
\put(32,0){$q$}


\put(11,11){\vector(1,1){1}}
\put(13,12){\vector(-1,0){1}}
\put(35.1,23){\vector(1,1){1}}
\put(15,0){$p$}
\put(1,8){1}
\put(13,24){2}
\put(30,24){3}
\put(19,16){\makebox(0,0){$q-3$}}
\end{picture}

\eb
 \[ {{V(q,{\it k3},-{\it k3} + q)\,
      W(p,-{\it k3} + q,{\it k1},{\it k2})}\over
    {-\omega ({\it k3}) - \omega ({\it k3} - q) + \omega (q)}} \]

   \ep

%picture77
%\input{pic73-80}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(12,11)(0,-4){3}{\oval(1.6,2)[l]}
\multiput(12,9)(0,-4){2}{\oval(1.6,2)[r]}

\put(12,12){\line(1,0){6}}
\multiput(19,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(21,12)(4,0){1}{\oval(2,1.6)[b]}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\put(36,0){\vector(-1,1){12}}
\put(12,22){\vector(-1,1){1}}
\put(2.2,12.8){\vector(-1,-1){1}}
\put(32,0){$q$}


\put(11,11){\vector(1,1){1}}
\put(23,13){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(15,0){$p$}
\put(1,8){2}
\put(13,24){3}
\put(30,24){1}
\put(19,16){\makebox(0,0){$1-q$}}
\end{picture}

\eb
 \[ {{G(p,{\it k2},{\it k3},{\it k1} - q)\,V({\it k1},q,{\it k1} - q)}\over
    {\omega ({\it k1}) - \omega (q) - \omega (-{\it k1} + q)}} \]

  \ep

%picture78
%\input{pic73-80}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(12,11)(0,-4){3}{\oval(1.6,2)[l]}
\multiput(12,9)(0,-4){2}{\oval(1.6,2)[r]}

\put(12,12){\line(1,0){6}}
\multiput(19,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(21,12)(4,0){1}{\oval(2,1.6)[b]}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\put(36,0){\vector(-1,1){12}}
\put(12,22){\vector(-1,1){1}}
\put(2.2,12.8){\vector(-1,-1){1}}
\put(32,0){$q$}


\put(11,11){\vector(1,1){1}}
\put(13,12){\vector(-1,0){1}}
\put(35.1,23){\vector(1,1){1}}
\put(15,0){$p$}
\put(1,8){2}
\put(13,24){3}
\put(30,24){1}
\put(19,16){\makebox(0,0){$q-1$}}
\end{picture}

\eb
 \[ {{V(q,{\it k1},-{\it k1} + q)\,
      W(p,-{\it k1} + q,{\it k2},{\it k3})}\over
    {-\omega ({\it k1}) + \omega (q) - \omega (-{\it k1} + q)}} \]

   \ep

%picture79
%4\input{pic73-80}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(12,11)(0,-4){3}{\oval(1.6,2)[l]}
\multiput(12,9)(0,-4){2}{\oval(1.6,2)[r]}

\put(12,12){\line(1,0){6}}
\multiput(19,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(21,12)(4,0){1}{\oval(2,1.6)[b]}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\put(36,0){\vector(-1,1){12}}
\put(12,22){\vector(-1,1){1}}
\put(2.2,12.8){\vector(-1,-1){1}}
\put(32,0){$q$}


\put(11,11){\vector(1,1){1}}
\put(23,13){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(15,0){$p$}
\put(1,8){1}
\put(13,24){3}
\put(30,24){2}
\put(19,16){\makebox(0,0){$2-q$}}
\end{picture}

\eb
 \[ {{G(p,{\it k1},{\it k3},{\it k2} - q)\,V({\it k2},q,{\it k2} - q)}\over
    {\omega ({\it k2}) - \omega ({\it k2} - q) - \omega (q)}} \]

   \ep

%picture80
%\input{pic73-80}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(12,11)(0,-4){3}{\oval(1.6,2)[l]}
\multiput(12,9)(0,-4){2}{\oval(1.6,2)[r]}

\put(12,12){\line(1,0){6}}
\multiput(19,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(21,12)(4,0){1}{\oval(2,1.6)[b]}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\put(36,0){\vector(-1,1){12}}
\put(12,22){\vector(-1,1){1}}
\put(2.2,12.8){\vector(-1,-1){1}}
\put(32,0){$q$}


\put(11,11){\vector(1,1){1}}
\put(13,12){\vector(-1,0){1}}
\put(35.1,23){\vector(1,1){1}}
\put(15,0){$p$}
\put(1,8){1}
\put(13,24){3}
\put(30,24){2}
\put(19,16){\makebox(0,0){$q-2$}}
\end{picture}

\eb
 \[ {{V(q,{\it k2},-{\it k2} + q)\,
      W(p,-{\it k2} + q,{\it k1},{\it k3})}\over
    {-\omega ({\it k2}) - \omega ({\it k2} - q) + \omega (q)}} \]
   \ep

 }





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\bibitem{Z92} Zakharov,V.E. Breaking Waves,
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\bibitem{Z74} Zakharov,V.E.,
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\bibitem{LZ} Lavrova,O.Yu. and Zakharov,V.E. (Unpublished)

\bibitem{ZSH} Zakharov,V.E., Shulman,E., {\em What is Integrability?\/},
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\end{thebibliography}
\end{document}





