Quantum $\mathfrak{sl}(n,\mathbb{C})$ link invariants
Myeong--Ju Jeong
Kuperberg generalized Temperley-Lieb algebras to the $\mathfrak{sl}(3)$ web spaces, we extend it to $\mathfrak{sl}(n)$ for $n\ge 4$. We find that the HOMFLY polynomial $P_n(q)$ can be computed by a linear expansion with respect to the relations of the quantum $\mathfrak{sl}(n)$ representation theory. We provide criterions which determine the periodicity of links by using the quantum $\mathfrak{sl}(n)$ HOMFLY polynomial.
math0506403.tex
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\begin{document}

\title{Quantum $\mathfrak{sl}(n,\mathbb{C})$ link invariants}

%    Information for first author
\author{Myeong--Ju Jeong}
%    Address of record for the research reported here
\address{Department of Mathematics, Korean Science Academy,
Pusan, 614-822, Korea} \email{determiner@hanmail.net}

\author{Dongseok Kim}
\address{Department of Mathematics, College of Natural Science,
Kyungpook National University, Taegu, 702-201, Korea}
\email{dongseok@knu.ac.kr}
\thanks{The first author was supported in
part by the Post-doctoral Fellowship Program of KOSEF and the second
author was supported in part by KRF Grant M02-2004-000-20044-0.}

\date{June 20, 2005, revised March 1, 2006}

\keywords{HOMFLY polynomial, Representation theory, Quantum special
linear algebras} \subjclass[2000]{Primary 57M27; Secondary 57M25,
57R56}


\begin{abstract}
Kuperberg generalized Temperley-Lieb algebras to the
$\mathfrak{sl}(3)$ web spaces, we extend it to $\mathfrak{sl}(n)$
for $n\ge 4$. We find that the HOMFLY polynomial $P_n(q)$ can be
computed by a linear expansion with respect to the relations of the
quantum $\mathfrak{sl}(n)$ representation theory. We provide
criterions which determine the periodicity of links by using the
quantum $\mathfrak{sl}(n)$ HOMFLY polynomial.
\end{abstract}

\maketitle

\section{Introduction}

The HOMFLY polynomial $P_n(q)$ can be calculated uniquely by the
following skein relations:

$$ P_{n}(\emptyset) =1, $$
$$ P_{n}( \pspicture[.1](-.17,-.17)(.17,.17) \pscircle(0,0){.15}
\endpspicture \cup D)= (\frac{q^{\frac n2}- q^{-\frac n2}}{q^{\frac 12}- q^{-\frac 12}}) P_{n}
(D),
$$
$$
q^{\frac n2}P_{n}(L_+) - q^{-\frac n2}P_{n}(L_-) = (q^{\frac 12}-
q^{-\frac 12}) P_n(L_0),
$$
where $\emptyset$ is the empty diagram,
$\pspicture[.2](-.17,-.17)(.17,.17) \pscircle(0,0){.15}
\endpspicture$ is the trivial knot and $L_+, L_-$ and $L_0$ are
three diagrams which are identical except at one crossing as in
Figure~\ref{local}~\cite{MOY:Homfly}. For each positive integer $n$,
the HOMFLY polynomial of links specializes to a one variable
polynomial that can be recovered from the quantum $\mathfrak{sl}(n)$
representation theory. For $n=0$, we use $\mathfrak{gl}(1|1)$ to
find that $P_0(q)$ is the Alexander polynomial~\cite{khvanov:comm}.
For $n=1$, one can see that $P_1(q)=1$ for all links. For $n=2$,
$P_2(q)$ is the Jones polynomial \cite{Jones:braid, RT:1, RT:2}. The
polynomial $P_n(q)$ can be computed by linearly expanding each
crossing into a sum of planar cubic graphs as in
Figure~\ref{planar}~\cite{MOY:Homfly}. The edges of these planar
cubic graphs are oriented and colored by $1$ or $2$.




In the language of the representation theory
\cite{Kuperberg:spiders, KR:factor}, the color $1$ of an edge
presents the vector representation $V$ of the quantum
$\mathfrak{sl}(n)$, and the color $2$ for its exterior power
$\wedge^2 V$. The trivalent vertex is the unique (up to scaling)
intertwiner of $V^{\otimes 2}\otimes \wedge^2 V$. This setup works
for arbitrary exterior powers of $V$: oriented edges of graphs carry
colors from $1$ to $n-1$ that denote the fundamental representations
of the quantum $\mathfrak{sl}(n)$ ~\cite{MOY:Homfly}. Kuperberg
generalized Temperley-Lieb algebras to $\mathfrak{sl}(3)$ web spaces
\cite{Kuperberg:spiders}, In section \ref{relation} we develop the
quantum $\mathfrak{sl}(n)$ representation theory by extending the
idea of webs in ~\cite{MOY:Homfly}. Sikora also found a
generalization of webs to the quantum $\mathfrak{sl}(n)$
\cite{Sikora:sln}. By expanding all crossings as in
Figures~\ref{pstvresol1} and~\ref{pstvresol2} Murakami, Ohtsuki and
Yamada found a regular isotopy invariant $[D]_n$~\cite{MOY:Homfly}.
In section~\ref{invariants} we modify writhes  suitably to define an
isotopy invariant $K_n(L,\mu)$ where $\mu$ is a coloring of $L$.
Using the quantum $\mathfrak{sl}(n)$ representation theory, we show
$K_n(L,\mu)$ can be computed by a linear expansion with respect to
the relations of web spaces in Theorem~\ref{onedim}.

\begin{figure}
\begin{eqnarray}\nonumber \pspicture[.4](-1,-1.5)(1,1)
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\rput(-.1,-.1){\rnode{b3}{$$}}
\rput(.1,-.1){\rnode{b4}{$$}}\ncline{a2}{b2}\middlearrow
\ncline{a1}{b1}\middlearrow \ncline{b4}{a4} \ncline{b1}{a3}
\rput(0,-1.2){\rnode{c4}{$L_+$}}
\endpspicture \hskip .2cm, \hskip .2cm
\pspicture[.4](-1,-1.5)(1,1) \rput(.7,.7){\rnode{a1}{$$}}
\rput(-.7,.7){\rnode{a2}{$$}} \rput(-.7,-.7){\rnode{a3}{$$}}
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\ncline{a1}{b1}\middlearrow \ncline{b2}{a4} \ncline{b3}{a3}
\rput(0,-1.2){\rnode{c4}{$L_-$}}
\endpspicture
\hskip .2cm , \hskip .2cm\pspicture[.4](-1,-1.5)(1,1)
\rput(.7,.7){\rnode{a1}{$$}} \rput(-.7,.7){\rnode{a2}{$$}}
\rput(-.7,-.7){\rnode{a3}{$$}} \rput(.7,-.7){\rnode{a4}{$$}}
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\rput(-.1,-.1){\rnode{b3}{$$}} \rput(.1,-.1){\rnode{b4}{$$}}
\nccurve[angleA=225,angleB=135]{a1}{a4}\middlearrow
\nccurve[angleA=315,angleB=45]{a2}{a3}\middlearrow
\rput(0,-1.2){\rnode{c4}{$L_0$}}
\endpspicture
\end{eqnarray}
\caption{The shape of $L_+, L_-$ and $L_0$.} \label{local}
\end{figure}

\begin{figure}
\begin{eqnarray}\nonumber
\pspicture[.55](-1.1,-1)(1.1,1) \rput(.7,.7){\rnode{a1}{$$}}
\rput(-.7,.7){\rnode{a2}{$$}} \rput(-.7,-.7){\rnode{a3}{$$}}
\rput(.7,-.7){\rnode{a4}{$$}} \rput(.1,.1){\rnode{b1}{$$}}
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\rput(.85,.8){\rnode{c1}{$1$}} \rput(-.85,.8){\rnode{c1}{$1$}}
\endpspicture &=& q^{\frac{1}{2}}
\pspicture[.55](-1.1,-1)(1.1,1)  \rput(.7,.7){\rnode{a1}{$$}}
\rput(-.7,.7){\rnode{a2}{$$}} \rput(-.7,-.7){\rnode{a3}{$$}}
\rput(.7,-.7){\rnode{a4}{$$}}
\nccurve[angleA=225,angleB=135]{a1}{a4}\middlearrow
\nccurve[angleA=315,angleB=45]{a2}{a3}\middlearrow
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\endpspicture -
\pspicture[.55](-1.1,-1)(1.1,1)  \rput(.7,.7){\rnode{a1}{$$}}
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\ncline{a2}{b1}\middlearrow \ncline{b1}{b2}\middlearrow
\ncline{b2}{a3}\middlearrow \ncline{b2}{a4}\middlearrow
\rput(.85,.8){\rnode{c1}{$1$}} \rput(-.85,.8){\rnode{c1}{$1$}}
\rput(-.85,-.8){\rnode{c1}{$1$}} \rput(.85,-.8){\rnode{c1}{$1$}}
\rput(.3,0){\rnode{c4}{$2$}}
\endpspicture \\\nonumber
\pspicture[.55](-1.1,-1)(1.1,1)  \rput(.7,.7){\rnode{a1}{$$}}
\rput(-.7,.7){\rnode{a2}{$$}} \rput(-.7,-.7){\rnode{a3}{$$}}
\rput(.7,-.7){\rnode{a4}{$$}} \rput(.1,.1){\rnode{b1}{$$}}
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\rput(.85,.8){\rnode{c1}{$1$}} \rput(-.85,.8){\rnode{c1}{$1$}}
\endpspicture &=& q^{-\frac{1}{2}}
\pspicture[.55](-1.1,-1)(1.1,1)  \rput(.7,.7){\rnode{a1}{$$}}
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\nccurve[angleA=315,angleB=45]{a2}{a3}\middlearrow
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\endpspicture -
\pspicture[.55](-1.1,-1)(1.1,1)  \rput(.7,.7){\rnode{a1}{$$}}
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\rput(0,-.3){\rnode{b2}{$$}} \ncline{a1}{b1}\middlearrow
\ncline{a2}{b1}\middlearrow \ncline{b1}{b2}\middlearrow
\ncline{b2}{a3}\middlearrow \ncline{b2}{a4}\middlearrow
\rput(.85,.8){\rnode{c1}{$1$}} \rput(-.85,.8){\rnode{c1}{$1$}}
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\rput(.3,0){\rnode{c4}{$2$}}
\endpspicture
\end{eqnarray}
\caption{Expansions of crossings for $P_n(q)$.} \label{planar}
\end{figure}


A link $L$ in $S^3$ is $p$-\emph{periodic} if there exists a
periodic homeomorphism $h$ of order $p$ such that $fix(h)\cong S^1,
h(L)=L$ and $fix(h)\cap L=\emptyset$ where $fix(h)$ is the set of
all fixed points of $h$. It is well known that if we consider $S^3$
as $\mathbb{R}\cup \{\infty\}$, we can assume that $h$ is a rotation
by $2\pi/p$ angle around the $z-$axis. If $L$ is a periodic link, we
denote its factor link by $\overline{L}$.
Murasugi~\cite{mu:alexander} found a strong relation between the
Alexander polynomials of a periodic link and its factor link.
Murasugi also found a similar relation for the Jones polynomials of
$L$ and $\overline L$~\cite{mu:jones}. Park and the first author
gave a criterion for periodic knots and lens knots by using
Vassiliev invariants \cite{JP5}. There are various result to decide
the periodicity of links \cite{Przytycki:criterion, PS:sun,
traczyk:period3, yokota:skein, yokota:kauffman, yokota:poly}. Chbili
generalized it to $P_3(q)$ using the quantum $\mathfrak{sl}(3)$
representation theory and stated a conjecture that the same
statement is true for $P_n(q)$ where $n\ge 2$ \cite{chbili}. In this
paper we find a positive answer after a slight modification.

\begin{theorem}
Let $p$ be a positive integer and $L$ be a $p-$periodic link in
$S^3$ with its factor link $\overline{L}$. Let $\mu$ be a
$p$-periodic coloring of $L$ and $\overline{\mu}$ be the induced
coloring of $\overline{L}$. Then for $n\ge 0$,

$$K_n(L,\mu) \cong K_n(\overline{L},\overline{\mu})^p \hskip ,7cm modulo \hskip .2cm
\mathcal{I}_n,$$ where $\mathcal{I}_n$ is the ideal of
$\mathbb{Z}[q^{\pm \frac 12}]$ generated by $p$ and
$\left[\begin{matrix}n\\
i\end{matrix}\right]^p-\left[\begin{matrix}n\\
i\end{matrix}\right]$ for $i=1, 2, \ldots, \lfloor \frac{n}{2}
\rfloor$. \label{mainthm}
\end{theorem}

The quantum integers are defined as
\begin{align*}
[n]&=\frac{q^{\frac{n}{2}}-q^{-\frac{n}{2}}}{q^{\frac{1}{2}}-q^{-\frac{1}{2}}},\\
 [n]!&= [n][n-1]\ldots [2][1], \\
\left[\begin{matrix} n \\ k \end{matrix}\right] &=
\frac{[n]!}{[n-k]![k]!}.
\end{align*}

Let us remark that even if we replace $K_n(L,\mu)$ by $P_n(q)$ in
Theorem~\ref{mainthm}, the ideal $\mathcal{I}_n$ remains the same.
Precisely, when we have a disjoint union of unknots after expanding
crossings and applying relations, the colors of unknots may not be
identically $1$ in general. Chen and Le found a similar result using
a different method~\cite{CL:period}, the improvements in Theorem
\ref{mainthm} are the followings: the result is for an isotopy link
invariant, $p$ does not have to be prime and the ideal runs only on
a finite set, the set of fundamental representations instead of an
infinite set, the set of all finite dimensional irreducible
representations of the quantum $\mathfrak{sl}(n)$.

We also give a criterion for periodic links by using the invariants
$K_n$ of a link and its mirror image in the following theorem.

\begin{theorem} \label{mainthm2}
\label{mirror2} Let $L$ be a $p$--periodic link and let $L^*$ be the
link obtained from the mirror image of a diagram of $L$. Let $\mu$
be a coloring of $L$ and $\mu^*$ be the coloring of $L^*$ induced
from the coloring $\mu$ of $L$. Then we have
$$K_n(L,\mu) \equiv K_n(L^*,\mu^*)  ~~\mathrm{mod}~~ (p, q^p-1).$$
\end{theorem}

In section~\ref{relation}, we develop the quantum $\mathfrak{sl}(n)$
representation theory. In section~\ref{invariants} we prove main
theorems and extend Theorem \ref{mainthm} to the colored
$\mathfrak{sl}(n)$ HOMFLY polynomial.

\section{The quantum $\mathfrak{sl}(n)$ representation theory}
\label{relation}




We refer to \cite{FultonHarris:gtm, Kuperberg:spiders, KRT:knot,
KR:factor} for the general representation theory. The webs are
generated by the two types of shapes in Figure~\ref{generator} where
$i+j+k=n$. At the second web in Figure~\ref{generator}, by using the
duality, $(V_{\lambda_i})^{*}\cong V_{\lambda_{n-i}}$, we can change
the directions of the edges to inward  and the colors of the
resulting edges to $n-i, n-j$ and $n-k$. After this process, we say
a vertex has a {\it sign type} $+$ if the sum of the colors is $n$,
$-$ if the sum is $2n$. The generators in Figure~\ref{generator}
have sign type $+, -$ from left. Naturally we can assign a sign type
for each face. Using relations~\ref{anfourid1} and~\ref{anfourid2}
in Figure~\ref{firstrel}, we only need to look at the face of a
\emph{valid sign type} in which $+, -$ are appearing alternatively.
One can easily see that none of faces with odd numbers of edges have
a valid sign type.

\begin{figure}
$$
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\rput(0,0){\rnode{b1}{$$}}\rput[bl](4pt,8pt){$k$}
\rput[bl](-13pt,-3pt){$i$} \rput[bl](13pt,-3pt){$j$}
\ncline{a1}{b1}\middlearrow \ncline{a2}{b1}\middlearrow
\ncline{a3}{b1}\middlearrow
\endpspicture
\hskip 1cm , \hskip 1cm \pspicture[.4](-1,-.6)(1,.9)
\rput(0,1){\rnode{a1}{$$}} \rput(-.866,-.5){\rnode{a2}{$$}}
\rput(.866,-.5){\rnode{a3}{$$}} \rput(0,0){\rnode{b1}{$$}}
\rput[bl](5pt,8pt){$k$} \rput[bl](-13pt,-3pt){$i$}
\rput[bl](13pt,-3pt){$j$}
 \ncline{b1}{a1}\middlearrow
\ncline{b1}{a2}\middlearrow \ncline{b1}{a3}\middlearrow
\endpspicture
$$
\caption{Generators of the quantum $\mathfrak{sl}(n)$ web space.}
\label{generator}
\end{figure}

\begin{figure}
\begin{eqnarray}
\pspicture[.4](-.6,-.3)(.6,.3) \pscircle(0,0){.4}
\rput[bl](14pt,0pt){$i$}
\endpspicture
& = & \left[ \begin{matrix} n\\
i \end{matrix} \right] \label{ancircle1}\\
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\rput(-1.2,0){\rnode{a1}{$$}}\rput(1.2,0){\rnode{a4}{$$}}
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\rput[al](0pt,13pt){$j$}
\nccurve[angleA=-120,angleB=-60]{a2}{a3}\middlearrow
\rput[al](-8pt,-15pt){$i-j$} \ncline{a3}{a4}\middlearrow
\endpspicture
& = &\left[ \begin{matrix} i\\ j \end{matrix}
\right]\pspicture[.45](-.8,-.8)(.8,.8) \rput(-.6,0){\rnode{a1}{$$}}
\rput(.6,0){\rnode{a2}{$$}} \rput(-.75,0){\rnode{b1}{$i$}}
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\pspicture[.45](-1.5,-1.1)(1.8,1) \rput(-.6,1.0){\rnode{c1}{$$}}
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\rput(-.8,-.9){\rnode{d2}{$j$}} \rput(.8,-.9){\rnode{d3}{$k$}}
\rput(1.4,.9){\rnode{d4}{$i+j+k$}} \rput(0,.4){\rnode{b1}{$$}}
\rput(0,-.4){\rnode{b2}{$$}} \ncline{c1}{b1}\middlearrow
\ncline{b2}{b1}\middlearrow \rput[bl](12pt,-3pt){$j+k$}
\ncline{c2}{b2}\middlearrow \ncline{c3}{b2}\middlearrow
\ncline{b1}{c4}\middlearrow
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& = & \pspicture[.45](-1.2,-1.1)(1.2,1)
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\ncline{c1}{b1}\middlearrow \ncline{b1}{b2}\middlearrow
\rput[bl](-11pt,5pt){$i+j$} \ncline{c2}{b1}\middlearrow
\ncline{c3}{b2}\middlearrow \ncline{b2}{c4}\middlearrow
\endpspicture \label{anfourid1}\\
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\rput(-.8,-.9){\rnode{d2}{$j$}}\rput(.8,-.9){\rnode{d3}{$k$}}
\rput(1.4,.9){\rnode{d4}{$i+j+k$}} \rput(0,.4){\rnode{b1}{$$}}
\rput(0,-.4){\rnode{b2}{$$}}\ncline{b1}{c1}\middlearrow
\ncline{b1}{b2}\middlearrow \rput[bl](12pt,-3pt){$j+k$}
\ncline{b2}{c2}\middlearrow \ncline{b2}{c3}\middlearrow
\ncline{c4}{b1}\middlearrow
\endpspicture
& = & \pspicture[.45](-1.2,-1)(1.2,1.1)
\rput(-1.0,.7){\rnode{c1}{$$}} \rput(-1.0,-.7){\rnode{c2}{$$}}
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\ncline{b1}{c1}\middlearrow \ncline{b2}{b1}\middlearrow
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\endpspicture \label{anfourid2}
\end{eqnarray}
\caption{Relations obtained by the quantum dimension and scaling.}
\label{firstrel}
\end{figure}

A few computations of tensor products of fundamental representations
show the relations in Figure~\ref{firstrel} (up to scaling): the
quantum dimension of the
fundamental representation $V_{\lambda_i}$ is $\left[ \begin{matrix} n\\
i \end{matrix} \right]$, the dimension of the invariant space of
$V_{\lambda_i} \otimes V_{\lambda_i}^*$ is $1$ and the dimension of
the invariant space of $V_{\lambda_i} \otimes V_{\lambda_j}\otimes
V_{\lambda_k} \otimes V_{\lambda_i+\lambda_j+\lambda_k}^*$ is $1$.


\subsection{Rectangular Relations}


\begin{figure}
$$
\{ \pspicture[.45](-1.5,-1.5)(1.5,1.5)\rput(.4,.4){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.4,.4){\rnode{a2}{$$}}
\rput(-.4,-.4){\rnode{a3}{$$}} \rput(.4,-.4){\rnode{a4}{$$}}
\rput(.8,.8){\rnode{b1}{$$}} \rput(-.8,.8){\rnode{b2}{$$}}
\rput(-.8,-.8){\rnode{b3}{$$}} \rput(.8,-.8){\rnode{b4}{$$}}
\rput(.9,.9){\rnode{d1}{$j$}} \rput(-.9,.9){\rnode{d2}{$i$}}
\rput(-.9,-.9){\rnode{d3}{$i$}} \rput(.9,-.9){\rnode{d4}{$j$}}
\rput(0pt,19pt){$l$} \rput(-25pt,0pt){$l+j$} \rput(0pt,-19pt){$l$}
\rput(25pt,0pt){$i-l$} \ncline{a1}{b1}\middlearrow
\ncline{a2}{b2}\middlearrow \ncline{b3}{a3}\middlearrow
\ncline{b4}{a4}\middlearrow \ncline{a1}{a2}\middlearrow
\ncline{a4}{a1}\middlearrow \ncline{a3}{a2}\middlearrow
\ncline{a3}{a4}\middlearrow
\endpspicture
\}_{l=0}^{i} \hskip .5cm \mathrm{or} \hskip .5cm \{
\pspicture[.45](-1.5,-1.5)(1.5,1.5)\rput(.4,.4){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.4,.4){\rnode{a2}{$$}}
\rput(-.4,-.4){\rnode{a3}{$$}} \rput(.4,-.4){\rnode{a4}{$$}}
\rput(.8,.8){\rnode{b1}{$$}} \rput(-.8,.8){\rnode{b2}{$$}}
\rput(-.8,-.8){\rnode{b3}{$$}} \rput(.8,-.8){\rnode{b4}{$$}}
\rput(.9,.9){\rnode{d1}{$j$}} \rput(-.9,.9){\rnode{d2}{$i$}}
\rput(-.9,-.9){\rnode{d3}{$i$}} \rput(.9,-.9){\rnode{d4}{$j$}}
\rput(0pt,19pt){$l$} \rput(-25pt,0pt){$i+l$} \rput(0pt,-19pt){$l$}
\rput(27pt,0pt){$j-l$} \ncline{a1}{b1}\middlearrow
\ncline{a2}{b2}\middlearrow \ncline{b3}{a3}\middlearrow
\ncline{b4}{a4}\middlearrow \ncline{a2}{a1}\middlearrow
\ncline{a4}{a1}\middlearrow \ncline{a3}{a2}\middlearrow
\ncline{a4}{a3}\middlearrow
\endpspicture
\}_{l=0}^{i}
$$
\caption{Two basis webs with the boundary $V_{\lambda_i}\otimes
V_{\lambda_j}\otimes V_{\lambda_i}^* \otimes V_{\lambda_j}^*$. }
\label{recbasis}
\end{figure}



To discuss rectangular relations, we first prove the following
lemma.

\begin{lemma}
\begin{enumerate}
\item If $i\le j\le n-j-1$, then
$$\mathrm{dim}(\mathrm{inv}(V_{\lambda_i}\otimes V_{\lambda_j}
\otimes V_{\lambda_i}^{*} \otimes V_{\lambda_j}^{*}))=i+1.$$
\item
If $j\ge i\ge k\ge 1, n-j-1\ge i$ and $n-i-j-1\ge l\ge 1$, then
$$\mathrm{dim}(\mathrm{inv}(V_{\lambda_i}\otimes V_{\lambda_{j+l}}
\otimes V_{\lambda_{i+l}}^{*} \otimes V_{\lambda_{j}}^{*}))=i+1.$$
\end{enumerate}
\label{twotensor}
\end{lemma}
\begin{proof}
If $i\le j\le n-j-1$, we obtain the following isomorphism by the
Clebsch-Gordan formula.

$$
V_{\lambda_i}\otimes V_{\lambda_j}\cong
V_{\lambda_i+\lambda_j}\oplus V_{\lambda_{i-1}+\lambda_{j+1}} \oplus
\cdots \oplus V_{\lambda_{1}+\lambda_{i+j}} \oplus
V_{\lambda_{i+j+1}}.$$

For irreducible representations $V, W$ of a simple Lie algebra, by a
simple application of Schur's lemma we find

$$\mathrm{dim}(\mathrm{Inv}( V\otimes W^*)) = \left\{
\begin{array}{cl} 1 & ~~\mathrm{if}~~ V\cong W, \\
0 & ~~\mathrm{if}~~ V\not\cong W. \end{array}\right.
$$

These two facts imply that
$\mathrm{dim}(\mathrm{inv}(V_{\lambda_i}\otimes V_{\lambda_j}
\otimes V_{\lambda_i}^{*} \otimes V_{\lambda_j}^{*}))=i+1.$
Similarly one can prove the other.
\end{proof}

\begin{figure}
\begin{eqnarray}
\pspicture[.45](-1.7,-1)(1.8,1.1) \rput(1.4,.3){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.8,.3){\rnode{a2}{$$}}
\rput(-.8,-.3){\rnode{a3}{$$}} \rput(1.4,-.3){\rnode{a4}{$$}}
\rput(1.7,.8){\rnode{b1}{$$}} \rput(-1.1,.8){\rnode{b2}{$$}}
\rput(-1.1,-.8){\rnode{b3}{$$}} \rput(1.7,-.8){\rnode{b4}{$$}}
\rput(2.3,.9){\rnode{c1}{$j+l$}} \rput(-1.3,.9){\rnode{c2}{$i$}}
\rput(-1.3,-.9){\rnode{c3}{$j$}} \rput(2.3,-.9){\rnode{c4}{$i+l$}}
\rput(7pt,20pt){$j+k-i$} \rput(-40pt,0pt){$j+k$}
\rput(7pt,-18pt){$k$} \rput(15pt,0pt){$i-k+l$}
\ncline{a1}{b1}\middlearrow \ncline{a2}{b2}\middlearrow
\ncline{b3}{a3}\middlearrow \ncline{b4}{a4}\middlearrow
\ncline{a2}{a1}\middlearrow \ncline{a4}{a1}\middlearrow
\ncline{a3}{a2}\middlearrow \ncline{a4}{a3}\middlearrow
\endpspicture
&=& \sum _{m=0}^{i}\left[ \begin{matrix} l\\
k-m \end{matrix} \right]\hskip .4cm
\pspicture[.45](-1.7,-1)(2,1.1)\rput(1.4,.3){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.8,.3){\rnode{a2}{$$}}
\rput(-.8,-.3){\rnode{a3}{$$}} \rput(1.4,-.3){\rnode{a4}{$$}}
\rput(1.7,.8){\rnode{b1}{$$}} \rput(-1.1,.8){\rnode{b2}{$$}}
\rput(-1.1,-.8){\rnode{b3}{$$}} \rput(1.7,-.8){\rnode{b4}{$$}}
\rput(2.3,.9){\rnode{c1}{$j+l$}} \rput(-1.3,.9){\rnode{c2}{$i$}}
\rput(-1.3,-.9){\rnode{c3}{$j$}} \rput(2.3,-.9){\rnode{c4}{$i+l$}}
\rput(7pt,20pt){$m$} \rput(-43pt,0pt){$i-m$}
\rput(7pt,-18pt){$j-i+m$} \rput(13pt,0pt){$j+l+m$}
\ncline{a1}{b1}\middlearrow \ncline{a2}{b2}\middlearrow
\ncline{b3}{a3}\middlearrow \ncline{b4}{a4}\middlearrow
\ncline{a1}{a2}\middlearrow \ncline{a4}{a1}\middlearrow
\ncline{a3}{a2}\middlearrow \ncline{a3}{a4}\middlearrow
\endpspicture \label{recrel1}
\end{eqnarray}
\begin{eqnarray}
\pspicture[.45](-1.7,-.9)(1.8,.9)\rput(1.4,.3){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.8,.3){\rnode{a2}{$$}}
\rput(-.8,-.3){\rnode{a3}{$$}} \rput(1.4,-.3){\rnode{a4}{$$}}
\rput(1.7,.8){\rnode{b1}{$$}} \rput(-1.1,.8){\rnode{b2}{$$}}
\rput(-1.1,-.8){\rnode{b3}{$$}} \rput(1.7,-.8){\rnode{b4}{$$}}
\rput(2.3,.9){\rnode{c1}{$j+l$}} \rput(-1.3,.9){\rnode{c2}{$i$}}
\rput(-1.3,-.9){\rnode{c3}{$j$}} \rput(2.3,-.9){\rnode{c4}{$i+l$}}
\rput(7pt,20pt){$k$} \rput(-40pt,0pt){$i+k$}
\rput(7pt,-18pt){$i-j+k$} \rput(14pt,0pt){$j-k+l$}
\ncline{a1}{b1}\middlearrow \ncline{a2}{b2}\middlearrow
\ncline{b3}{a3}\middlearrow \ncline{b4}{a4}\middlearrow
\ncline{a2}{a1}\middlearrow \ncline{a4}{a1}\middlearrow
\ncline{a3}{a2}\middlearrow \ncline{a4}{a3}\middlearrow
\endpspicture
&=& \sum _{m=0}^{j}\left[ \begin{matrix} l\\
k-m \end{matrix} \right]\hskip .4cm
\pspicture[.45](-1.7,-.9)(2,.9)\rput(1.4,.3){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.8,.3){\rnode{a2}{$$}}
\rput(-.8,-.3){\rnode{a3}{$$}} \rput(1.4,-.3){\rnode{a4}{$$}}
\rput(1.7,.8){\rnode{b1}{$$}} \rput(-1.1,.8){\rnode{b2}{$$}}
\rput(-1.1,-.8){\rnode{b3}{$$}} \rput(1.7,-.8){\rnode{b4}{$$}}
\rput(2.3,.9){\rnode{c1}{$j+l$}} \rput(-1.3,.9){\rnode{c2}{$i$}}
\rput(-1.3,-.9){\rnode{c3}{$j$}} \rput(2.3,-.9){\rnode{c4}{$i+l$}}
\rput[a0](7pt,20pt){$i-j+m$} \rput[a0](-43pt,0pt){$j-m$}
\rput[a0](7pt,-18pt){$m$} \rput[a0](13pt,0pt){$i+l+m$}
\ncline{a1}{b1}\middlearrow \ncline{a2}{b2}\middlearrow
\ncline{b3}{a3}\middlearrow \ncline{b4}{a4}\middlearrow
\ncline{a1}{a2}\middlearrow \ncline{a4}{a1}\middlearrow
\ncline{a3}{a2}\middlearrow \ncline{a3}{a4}\middlearrow
\endpspicture \label{recrel2}
\end{eqnarray}
\caption{ Equations in Lemma \ref{rectangular}.} \label{recrel}
\end{figure}



\begin{figure}
$$
\pspicture[.45](-1.4,-.8)(1.5,.8) \rput(-1.0,.7){\rnode{c1}{$$}}
\rput(-1.0,-.7){\rnode{c2}{$$}} \rput(1.0,-.7){\rnode{c3}{$$}}
\rput(1.0,.7){\rnode{c4}{$$}} \rput(-1.1,.8){\rnode{d1}{$i$}}
\rput(-1.1,-.8){\rnode{d2}{$j$}} \rput(1.1,-.8){\rnode{d3}{$i+l$}}
\rput(1.1,.8){\rnode{d4}{$j+l$}} \rput(-.4,0){\rnode{b1}{$$}}
\rput(.4,0){\rnode{b2}{$$}} \ncline{b1}{c1}\middlearrow
\ncline{b1}{b2}\middlearrow \rput[b](2pt,5pt){$j-i$}
\ncline{c2}{b1}\middlearrow \ncline{b2}{c4}\middlearrow
\ncline{c3}{b2}\middlearrow
\endpspicture
=  \left[ \begin{matrix} l\\
0 \end{matrix} \right] \pspicture[.45](-1.4,-.8)(1.5,.8)
\rput(-1.0,.7){\rnode{c1}{$$}}
\rput(-1.0,-.7){\rnode{c2}{$$}}\rput(1.0,-.7){\rnode{c3}{$$}}
\rput(1.0,.7){\rnode{c4}{$$}} \rput(-1.1,.8){\rnode{d1}{$i$}}
\rput(-1.1,-.8){\rnode{d2}{$j$}} \rput(1.1,-.8){\rnode{d3}{$i+l$}}
\rput(1.1,.8){\rnode{d4}{$j+l$}}\rput(-.4,0){\rnode{b1}{$$}}
\rput(.4,0){\rnode{b2}{$$}} \ncline{b1}{c1}\middlearrow
\ncline{b1}{b2}\middlearrow \rput[b1](2pt,5pt){$j-i$}
\ncline{c2}{b1}\middlearrow \ncline{b2}{c4}\middlearrow
\ncline{c3}{b2}\middlearrow
\endpspicture
$$
\caption{Initial induction step $k=0$ and $j\ge i$.}
\label{induction1}
\end{figure}


\begin{figure}
$$
\pspicture[.45](-1.5,-1.3)(1.9,1.3)\rput(.5,.5){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.5,.5){\rnode{a2}{$$}}
\rput(-.5,-.5){\rnode{a3}{$$}} \rput(.5,-.5){\rnode{a4}{$$}}
\rput(1,1){\rnode{b1}{$$}} \rput(-1,-1){\rnode{b3}{$$}}
\rput(1,-1){\rnode{b4}{$$}} \rput(1.1,1.1){\rnode{c1}{$j+l$}}
\rput(-1.1,-1.1){\rnode{c3}{$j$}} \rput(1.1,-1.1){\rnode{c4}{$l$}}
\rput[a0](-32pt,0pt){$j+k$} \rput[a0](0pt,-25pt){$k$}
\rput[a0](37pt,0pt){$l-k$} \ncline{a1}{b1}\middlearrow
\ncline{b3}{a3}\middlearrow \ncline{b4}{a4}\middlearrow
\ncline{a4}{a1}\middlearrow
\nccurve[angleA=90,angleB=180]{a3}{a1}\middlearrow
\ncline{a4}{a3}\middlearrow
\endpspicture
= \pspicture[.45](-1.2,-1.3)(2,1.3)\rput(.5,.5){\rnode{a1}{$$}}
\rput(-.5,.5){\rnode{a0}{$$}} \rput(0,0){\rnode{a40}{$$}}
\rput(.5,-.5){\rnode{a43}{$$}} \rput(1,1){\rnode{b1}{$$}}
\rput(-1,-1){\rnode{b3}{$$}} \rput(1,-1){\rnode{a44}{$$}}
\rput(1.1,1.3){\rnode{c1}{$j+l$}} \rput(-1.2,-1.1){\rnode{c3}{$j$}}
\rput(1.1,-1.1){\rnode{c44}{$l$}} \rput[a0](5pt,-20pt){$k$}
\rput[a0](30pt,-5pt){$l-k$}
\nccurve[angleA=45,angleB=180]{a0}{b1}\middlearrow
\nccurve[angleA=90,angleB=225]{b3}{a0}\middlearrow
\ncline{a44}{a43}\middlearrow \ncline{a40}{a0}\middlearrow
\nccurve[angleA=45,angleB=45]{a43}{a40}\middlearrow
\nccurve[angleA=225,angleB=225]{a43}{a40}\middlearrow
\endpspicture
= \left[ \begin{matrix} l\\
k \end{matrix} \right]
\pspicture[.45](-1.2,-1.3)(2,1.3)\rput(.5,.5){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(1,1){\rnode{b1}{$$}}
\rput(-1,-1){\rnode{b3}{$$}} \rput(1,-1){\rnode{a44}{$$}}
\rput(1.1,1.1){\rnode{c1}{$j+l$}} \rput(-1.1,-1.1){\rnode{c3}{$j$}}
\rput(1.1,-1.1){\rnode{c44}{$l$}} \ncline{a0}{b1}\middlearrow
\ncline{b3}{a0}\middlearrow \ncline{a44}{a0}\middlearrow
\endpspicture
$$
\caption{Initial induction step $i=0$.} \label{induction2}
\end{figure}

From Lemma~\ref{twotensor}, we know the number of basis webs that we
need for each expansion. For $n-i-1\ge j\ge i \ge 0$, we can have
two sets of basis webs and each has the same sign type as in Figure
\ref{recbasis}. Throughout the section we will use the basis in the
left hand side of Figure~\ref{recbasis}. There are only two possible
types of rectangular relations as in Lemma ~\ref{rectangular}, all
other can be taken care of by relations~\ref{anfourid1} and
\ref{anfourid2} in Figure~\ref{firstrel}. The equation \ref{recrel1}
in Lemma \ref{rectangular} was appeared in \cite{MOY:Homfly} without
a proof.



\begin{figure}
$$ \pspicture[.45](-1.3,-.2)(1.3,1)\rput(.5,.5){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.5,.5){\rnode{a2}{$$}}
\rput(-.5,-.2){\rnode{a3}{$$}} \rput(.5,-.2){\rnode{a4}{$$}}
\rput(1,1){\rnode{b1}{$$}} \rput(-1,1){\rnode{b2}{$$}}
\rput(1.1,1.1){\rnode{c1}{$j+l+s$}}
\rput(-1.1,1.1){\rnode{c2}{$i-s$}} \rput(-1,-1){\rnode{b3}{$$}}
\rput(1,-1){\rnode{b4}{$$}} \rput[a0](0pt,25pt){$s$}
\rput[a0](-25pt,0pt){$i$} \rput[a0](37pt,0pt){$j+l$}
\ncline{a1}{b1}\middlearrow \ncline{a2}{b2}\middlearrow
\ncline{a2}{a1}\middlearrow \ncline{a4}{a1}\middlearrow
\ncline{a3}{a2}\middlearrow
\endpspicture
$$
\caption{} \label{Hs}
\end{figure}

\begin{lemma}
For $n-i-1 \ge j\ge i\ge k\ge 0$ and $n-i-j-1\ge l\ge 0$, we find
equation~\ref{recrel1} in Figure~\ref{recrel}. For $n-j-1 \ge i\ge
j\ge k\ge 1$ and $n-i-j-1\ge l\ge 1$, we find equation~\ref{recrel2}
in Figure~\ref{recrel}. Just for these two equations, we use a
different convention of quantum integers that
$\left[ \begin{matrix} 0\\
0 \end{matrix} \right]=1$ but $\left[ \begin{matrix} 0\\
k \end{matrix} \right]=0$ if $k \neq 0$. \label{rectangular}
\end{lemma}

\begin{proof}


\begin{figure}
\begin{eqnarray}
\pspicture[.45](-1.2,-1.3)(2.7,2.5)
\psline(0,0)(0,.8)\psline[arrowscale=1.5]{->}(0,.3)(0,.5)
\psline(0,.8)(0,1.6)\psline[arrowscale=1.5]{->}(0,1.1)(0,1.3)
\psline(0,0)(1,0)\psline[arrowscale=1.5]{->}(.6,0)(.4,0)
\psline(0,.8)(1,.8)\psline[arrowscale=1.5]{->}(.4,.8)(.6,.8)
\psline(0,1.6)(1,1.6)\psline[arrowscale=1.5]{->}(.4,1.6)(.6,1.6)
\psline(1,0)(1,1)\psline[arrowscale=1.5]{->}(1,.3)(1,.5)
\psline(1,.8)(1,1.6)\psline[arrowscale=1.5]{->}(1,1.1)(1,1.3)
\psline(0,0)(-.5,-.5)\psline[arrowscale=1.5]{->}(-.4,-.4)(-.2,-.2)
\psline(1,0)(1.5,-.5)\psline[arrowscale=1.5]{->}(1.4,-.4)(1.2,-.2)
\psline(0,1.6)(-.5,2.1)\psline[arrowscale=1.5]{->}(-.2,1.8)(-.4,2)
\psline(1,1.6)(1.5,2.1)\psline[arrowscale=1.5]{->}(1.2,1.8)(1.4,2)
\rput[br](-.2,.3){$j+k$} \rput[bl](-.2,1.1){$i$}
\rput[bl](1.2,.3){$i-k+l$} \rput[bl](1.2,1.1){$j+l$}
\rput[bl](.4,-.4){$k$} \rput[bl](.4,1.8){$1$}
\rput[b](-.6,2.2){$i-1$} \rput[b](1.6,2.2){$j+l+1$}
\rput[t](-.6,-.6){$j$} \rput[t](1.6,-.6){$i+l$}
\endpspicture
=\left[ \begin{matrix} j+k-i+1\\
1 \end{matrix} \right] \pspicture[.4](-1.2,-1.1)(2.7,2.6)
\psline(0,0)(0,1.5)\psline[arrowscale=1.5]{->}(0,.6)(0,.8)
\psline(1.5,0)(1.5,1.5)\psline[arrowscale=1.5]{->}(1.5,.6)(1.5,.8)
\psline(0,0)(1.5,0)\psline[arrowscale=1.5]{->}(.8,0)(.6,0)
\psline(0,1.5)(1.5,1.5)\psline[arrowscale=1.5]{->}(.6,1.5)(.8,1.5)
\psline(0,0)(-.5,-.5)\psline[arrowscale=1.5]{->}(-.4,-.4)(-.2,-.2)
\psline(1.5,0)(2,-.5)\psline[arrowscale=1.5]{->}(1.9,-.4)(1.7,-.2)
\psline(0,1.5)(-.5,2)\psline[arrowscale=1.5]{->}(-.2,1.7)(-.4,1.9)
\psline(1.5,1.5)(2,2)\psline[arrowscale=1.5]{->}(1.7,1.7)(1.9,1.9)
\rput[b](-.6,2.1){$i-1$} \rput[b](2.1,2.1){$j+l+1$}
\rput[t](-.6,-.6){$j$} \rput[t](2.1,-.6){$i+l$}
\rput[br](-.2,.7){$j+k$} \rput[bl](1.7,.7){$i-k+l$}
\rput[bl](.7,-.4){$k$}
\endpspicture \nonumber\\ \nonumber
=\left[ \begin{matrix} j+k-i+1\\
1 \end{matrix} \right] \sum_{m=0}^{i-1} \left[ \begin{matrix} l+1\\
k-m \end{matrix} \right] \pspicture[.45](-1.9,-.7)(2.7,1.5)
\psline(0,0)(0,.8)\psline[arrowscale=1.5]{->}(0,.4)(0,.6)
\psline(1,0)(1,.8)\psline[arrowscale=1.5]{->}(1,.4)(1,.6)
\psline(0,0)(1,0)\psline[arrowscale=1.5]{->}(.4,0)(.6,0)
\psline(0,.8)(1,.8)\psline[arrowscale=1.5]{->}(.6,.8)(.4,.8)
\psline(0,0)(-.5,-.5)\psline[arrowscale=1.5]{->}(-.4,-.4)(-.2,-.2)
\psline(1,0)(1.5,-.5)\psline[arrowscale=1.5]{->}(1.4,-.4)(1.2,-.2)
\psline(0,.8)(-.5,1.3)\psline[arrowscale=1.5]{->}(-.2,1)(-.4,1.2)
\psline(1,.8)(1.5,1.3)\psline[arrowscale=1.5]{->}(1.2,1)(1.4,1.2)
\rput[br](-.6,1.3){$i-1$} \rput[bl](1.6,1.3){$j+l+1$}
\rput[tr](-.6,-.6){$j$} \rput[tl](1.6,-.6){$i+l$}
\rput[br](-.2,.2){$i-m-1$} \rput[bl](1.2,.2 ){$j+l+m+1$}
\rput[bl](.7,1){$m$}
\endpspicture
\end{eqnarray}
\caption{} \label{lhs1}
\end{figure}

Let $a(i,m)$ be the coefficients in the righthand side of the
equations. We induct on $(\mathrm{min}\{i,j\},k)$ in lexicographic
order. The key idea is to prove both equations simultaneously. If
$k=0$ and $j\ge i$, we find the equation in Figure~\ref{induction1}.
For the case $k=0$ and $i\ge j$, it is identical except the weight
on the horizontal arrow is replaced by the weight $i-j$ of the
opposite direction. For $i=0$, we find the equations in
Figure~\ref{induction2}. One can do for the case $j=0$ similarly.
\end{proof}


Now we proceed to the induction step. Let us look at the case $n-i-1
\ge j\ge i\ge k\ge 0$, $n-i-j-1\ge l\ge 0$. On the top of each web
in equations~\ref{recrel1} and~\ref{recrel2} in Figure \ref{recrel},
we can attach $i$ different $H$'s, given in Figure~\ref{Hs} where
$i\ge s\ge 1$. If $s=i$, we can easily get

\begin{figure}
{\small
$$
\pspicture[.45](-1.2,-.6)(3.3,2)
\psline(0,0)(0,.8)\psline[arrowscale=1.5]{->}(0,.3)(0,.5)
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\caption{} \label{rhs1}
\end{figure}


$$
\left[ \begin{matrix} j+k\\
i \end{matrix} \right] \left[ \begin{matrix} i+l\\
k \end{matrix} \right] = \sum_{m=0}^i a(i,m) \left[ \begin{matrix} j+l+m\\
m \end{matrix} \right] \left[ \begin{matrix} j\\
i-m \end{matrix} \right].
$$

From the case $s=1$, we find that the left hand side of equation
\ref{recrel2} can be changed as the equations in Figure~\ref{lhs1}.
We use the induction hypothesis to find that each term in right hand
side of equation~\ref{recrel2} can be transformed as the equations
in Figure~\ref{rhs1} where $\alpha=[-i+j+l+m+1]$, $\beta=1$ and
$\gamma=[-i+j+m+1]$. By comparing coefficients of each basis
element, we get $i$ equations




$$
[-i+j+k+1]\left[ \begin{matrix} l+1\\
k-t \end{matrix} \right] = a(i,t) [-i+j+t+1]+a(i,t+1) [-i+j+l+t+2],
$$
where $i-1 \ge t \ge 0$. Since these $i+1$ equations are
independent, we plug in the answer to equations to check these are
correct. One can follow the proof for the second case.


\subsection{The quantum $\mathfrak{sl}(n)$ skein modules}

The authors have been trying to find a complete relation of the
quantum  of $\mathfrak{sl}(n)$ representation theory, but we find
that the bigger $n$, the bigger the size of a polygon we have to
find a suitable relation: there are a rectangular relation for the
quantum  of $\mathfrak{sl}(3)$, a hexagonal relation for the quantum
of $\mathfrak{sl}(4)$ and an octagonal relation for the quantum  of
$\mathfrak{sl}(6)$. In particular, it was conjectured a complete set
of relations for the quantum $\mathfrak{sl}(4)$ representation
theory \cite{Dongseok:thesis}. For large $n$, we find the following
remark.

\begin{remark}
We conjecture that any $2n$ polygon of the sign type $(+,$ $-,$ $+,$
$\ldots,$ $-)$ can be expanded to a sum of webs of polygons of
smaller sizes and $2n$ polygons of the sign type $(-$, $+$, $-$,
$\ldots$, $+)$.
\end{remark}

Somehow the study of the quantum $ \mathfrak{sl}(n)$
representation theory is not complete yet, but we can prove the
following theorem.

\begin{theorem} The quantum $\mathfrak{sl}(n)$ skein module of the plane or sphere
has dimension $1$. \label{onedim}
\end{theorem}

\begin{proof}

If we look at these webs without decorations, they are directed,
weighted, trivalent planar graphs. We will consider these graphs are
on $\mathbb{S}^2$ instead of $\mathbb{R}^2$. Let $V, E$ and $F$ be
the number of vertices, edges and faces in a graph on
$\mathbb{S}^2$, respectively. Let $F_i$ be the number of faces with
$i$ edges. We can easily find the following equations:

\begin{eqnarray}\nonumber
&3V =2E\\\nonumber
&2=V-E+F \\
&2E= \sum_{i} iF_i \label{graphformula1}
\end{eqnarray}

Throughout the section all webs are assumed to be without boundary
unless we state differently. We will show the dimension of the space
of webs without boundary is $1$. Suppose it is not true, then there
exists a web which is not an element in $\mathbb{C} [q^{\frac12},
q^{-\frac12}]$, say $D$. We assume that $D$ has the smallest number
of faces among all counterexamples. Since a face of size less than
$4$ can be removed by using relations we found in
Figure~\ref{firstrel}, we will assume that all faces in $D$ have at
least $4$ edges and we further assume that $D$ has the maximum
number of rectangular faces.


\begin{lemma}
If there exists a pentagonal face in $D$, it is adjacent to a unique
rectangular face. \label{a3lem1}
\end{lemma}

\begin{proof}
It is easy to see that there are no two adjacent rectangular faces
by considering Figure \ref{recbasis}. Thus if a pentagon is not
adjacent to a rectangle, the sizes of all adjacent faces of the
pentagon are bigger than $5$ because an adjoint pentagon can be
changed to a rectangle. By applying relations~\ref{anfourid1} and
\ref{anfourid2} in Figure \ref{firstrel} to the pentagonal face, it
can be changed to a rectangle without changing the total number of
faces in $D$ but increasing the number of the rectangles by $1$.
This contradicts the maximality of the number of the rectangular
faces. Therefore, it must be adjacent to a rectangle. There are five
possible shapes which can be obtained from the web in
Figure~\ref{penta} by $\frac{2\pi}{5}$ rotation. By applying
equations in Lemma \ref{rectangular}, one can see that any two out
of these five shapes are related as the equation in Figure
~\ref{pentarel}. Therefore, the adjacent rectangle is unique.
\end{proof}



A rectangular face is called \emph{isolated} if the sizes of all
adjacent faces (after making valid sign types) are bigger than $5$.
Otherwise, it is adjacent to the unique pentagon by Lemma
\ref{a3lem1}. Now we will look at the neighborhood of each isolated
rectangular face in subsection~\ref{reciso} and obtain the following
inequality.

\begin{eqnarray}\nonumber
4F_4 &\le 3F_7 + 4F_8 + 4F_9+5F_{10}+5F_{11}+ \ldots + \lfloor
\frac{n}{2} \rfloor F_n + \ldots, \\ \nonumber 5F_4 &\le 4F_7 + 5F_8
+ 5F_9 + 7 F_{10} + 7F_{11} + \ldots + n F_n + \ldots ,
\end{eqnarray}
for $n\ge 12$.

By looking at the neighborhood of a pentagon adjacent to a unique
rectangle, we find the following inequality in
subsection~\ref{recpen}.

\begin{figure}
$$
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\endpspicture $$
\caption{A valid shapes of a pentagon face.} \label{penta}
\end{figure}

\begin{figure}
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\endpspicture + \sum_i \beta_i ~~~\begin{matrix}\mathrm{webs}\hskip .2cm \mathrm{which}\hskip .2cm
\mathrm{have}\hskip .2cm \\ \mathrm{at}\hskip .2cm
\mathrm{most}\hskip .2cm \mathrm{one}\hskip .2cm
\mathrm{face}\end{matrix}
$$
\caption{Pentagonal relations.} \label{pentarel}
\end{figure}

$$
5(F_5+F_4) \le 4F_8+ 4F_9+5F_{10}+5F_{11}+ \ldots + \lfloor
\frac{n}{2} \rfloor F_n + \ldots .
$$

Now let us proceed to the proof. By combining these two inequalities
and the inequality, $(n-6) \ge \frac{2}{5}n$ for all $n\ge 12$, we
get the following inequality.


\begin{eqnarray}\nonumber
5F_5+10F_4&\le 3F_7+ 9F_8+ 9F_9+12F_{10} + 12F_{11} \\ \nonumber
&+ \ldots + (n +\lfloor \frac{n}{2} \rfloor) F_n + \ldots,\\
\nonumber 2F_4 +F_5 &\le 1F_7+2F_8+ 3F_9+4F_{10}+5F_{11}
\\&+\ldots + (n-6)F_n + \ldots .
\end{eqnarray}

Then,
\begin{eqnarray}\nonumber
6F &= 6F_4+6F_5+6F_6+6F_7+6F_8+ \ldots + 6F_n + \ldots \\ \nonumber
 &\le 4F_4+5F_5+6F_6+7F_7+8F_8+ \ldots + nF_n + \ldots = 2E.
\end{eqnarray}

If we plug this inequality into equations \ref{graphformula1}, we
find a contradiction as

$$2=V-E+F  \le \frac{2}{3}E -E + \frac{1}{3}E =0.$$

Therefore, this completes the proof.

\end{proof}

\subsubsection{Isolated rectangular faces}
\label{reciso}



From Lemma \ref{a3lem1}, we know all adjacent faces are either
hexagons or bigger polygons. Moreover, all adjacent hexagonal faces
must have a valid sign type (otherwise, they become two adjacent
rectangular faces). We will divide cases with the number of adjacent
hexagonal faces. If there is no adjacent hexagonal face, then it is
surrounded by at least four faces of sizes bigger than or equal to
$8$ and

$$4F_4 \le 4 F_8+ 4 F_9 + \ldots + \lfloor \frac{n}{2} \rfloor F_n + \ldots .$$


\begin{figure}
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$$
\caption{Just one hexagon in the neighborhood of the
rectangle.}\label{hexa3}
\end{figure}


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\qline(-3.964,-.5)(-3.964,.5) \psline(-3.964,.5)(-4.83,1)
\psline(-3.964,-.5)(-4.83,-1) \psline(.5,.5)(1.366,1)
\psline(.5,-.5)(1.366,-1)
\qline(1.366,1)(1.366,1.5)\qline(1.366,-1)(1.366,-1.5)
\qline(1.366,1)(2.232,.5)\qline(1.366,-1)(2.232,-.5)
\qline(2.232,-.5)(2.232,.5) \psline(2.232,.5)(2.665,.75)
\psline(2.232,-.5)(2.665,-.75)
\qline(3.866,.5)(3.433,.75)\qline(3.866,-.5)(3.433,-.75)
\qline(3.866,.5)(3.866,-.5) \psline(3.866,.5)(4.732,1)
\psline(3.866,-.5)(4.732,-1)
\qline(4.732,1)(4.732,1.5)\qline(4.732,-1)(4.732,-1.5)
\qline(4.732,1)(5.598,.5)\qline(4.732,-1)(5.598,-.5)
\qline(5.598,-.5)(5.598,.5) \psline(5.598,.5)(6.464,1)
\psline(5.598,-.5)(6.464,-1)
\endpspicture
$$
\caption{Two non-adjacent hexagons in the neighborhood of the
rectangle. } \label{hexa1}
\end{figure}

If there is just one hexagonal face, Lemma~\ref{rectangular} allows
us to change a rectangular face to the sum of rectangular (or less)
faces with the opposite sign type, this forces a change which is
called a \emph{swapping} of the rectangle and the hexagon as in
Figure~\ref{hexa3} without any restrictions on the sizes of neighbor
faces. If there is only one hexagon in the neighborhood of the
rectangle after a swapping, then we find the desired inequality as
the sizes of faces indicated by numbers in Figure~\ref{hexa3}. If
there are more than one hexagon, it will be dealt in the next cases.


If there are two adjacent hexagonal faces, there are also two
possibilities: If these two hexagons are adjacent, the rectangular
face can be changed into two rectangles which contradicts the
hypothesis that $D$ has the maximum number of rectangles as in
Figure \ref{hexa2}. If these two hexagons are not adjacent each
other, then we can continue to swap and we repeat the process at
the new rectangle. Since there are only finitely many faces, it
stops at finite steps. By tracing the swap moves, we find that it
is just like in Figure~\ref{hexa1}. For all other cases, we must
have at least two adjacent hexagons. However, it can be expanded
into a sum of webs which has either less faces or more rectangular
faces as shown in Figure \ref{hexa2} which contradict the
hypothesis of $D$.



\begin{figure}
$$
\pspicture[.45](-3.2,-2.2)(2.2,2) \rput(1.5,0){\rnode{e1}{$7^+$}}
\rput(0,1.5){\rnode{e3}{$7^+$}} \rput(-1.5,1.5){\rnode{e4}{$6^+$}}
\rput(-2,0){\rnode{e5}{$6^+$}} \rput(-1.5,-1.5){\rnode{e6}{$6^+$}}
\rput(0,-2){\rnode{e7}{$6^+$}} \rput(1.5,-1.5){\rnode{e8}{$6^+$}}
\qline(-.5,-.5)(-.5,.5) \qline(-.5,-.5)(.5,-.5)
\qline(.5,.5)(-.5,.5) \qline(.5,.5)(.5,-.5)
\qline(-1.5,-.5)(-1.5,.5) \qline(-.5,-1.5)(.5,-1.5) \qline(1,1)(1,2)
\qline(1,1)(2,1) \qline(-1,-1)(-1.5,-.5) \qline(-1,-1)(-.5,-1.5)
\qline(1,-1)(2,-1) \qline(1,-1)(.5,-1.5) \qline(-1,1)(-1.5,.5)
\qline(-1,1)(-1,2) \psline(.5,.5)(1,1) \psline(-.5,.5)(-1,1)
\psline(.5,-.5)(1,-1) \psline(-.5,-.5)(-1,-1) \psline(.5,-1.5)(1,-2)
\psline(-.5,-1.5)(-1,-2) \psline(-1.5,.5)(-2,1)
\psline(-1.5,-.5)(-2,-1)
\endpspicture
= \pspicture[.45](-3.2,-2.2)(3.2,2)
\rput(0,0){\rnode{e1}{$7$}}\rput(2,.5){\rnode{e2}{$6^+$}}
\rput(.5,2){\rnode{e3}{$6^+$}}\rput(-1.5,1.5){\rnode{e4}{$7^+$}}
\rput(-2.5,0){\rnode{e5}{$6^+$}}\rput(-1.5,-1.5){\rnode{e6}{$7^+$}}
\rput(0,-2.5){\rnode{e7}{$6^+$}}\rput(1.5,-1.5){\rnode{e8}{$7^+$}}
\psline(1,1)(1.5,1.5) \psline(-.5,1)(-1,.5) \psline(-2.5,1)(-2,.5)
\psline(-2.5,-1)(-2,-.5) \psline(-.5,-1)(-1,-.5)
\psline(-.5,-2)(-1,-2.5) \psline(.5,-2)(1,-2.5)
\psline(.5,-1)(1,-.5) \qline(1.5,1.5)(1.5,2)\qline(1.5,1.5)(2,1.5)
\qline(-.5,2.2)(-.5,1) \qline(-.5,1)(1,1) \qline(1,1)(1,-.5)
\qline(1,-.5)(2.5,-.5) \qline(-1,.5)(-2,.5) \qline(-1,.5)(-1,-.5)
\qline(-2,-.5)(-2,.5) \qline(-2,-.5)(-1,-.5) \qline(.5,-1)(.5,-2)
\qline(.5,-1)(-.5,-1) \qline(-.5,-2)(.5,-2) \qline(-.5,-2)(-.5,-1)
\endpspicture
$$
\caption{Two adjacent hexagons in the neighborhood of the rectangle.
} \label{hexa2}
\end{figure}

As we have seen, for each rectangular face, there are at least four
polygons which have more than seven faces. It is easy to see that
for each face of size $n\ge 7$, there are at most $\frac{n}{2}$ (or
$\frac{n-3}{2})$ rectangular faces which can travel by swapping
moves to the given face if $n$ is even (or odd, respectively).
Therefore, we get the inequality we desired.

\subsubsection{Pentagonal face with a unique adjacent rectangular
faces} \label{recpen}

We will treat these two adjacent polygons as a pentagon in this
subsection.

\begin{lemma}
The size of all adjacent polygons must be bigger than $7$.
\label{neighpenta}
\end{lemma}

\begin{proof}
First we already know these polygons can be neither rectangles nor
pentagons. Suppose one of them is either a hexagon or a heptagon.
Either case, we transform the shapes of these polygons that the
rectangular face is adjacent to the hexagon (now rectangle has a
valid sign type, we can change the heptagon to a hexagon). By the
relation in Figure \ref{pentarel}, we assume the hexagonal face is
adjacent to the rectangular face. Then as we have seen in Figure
\ref{hexa3}, the rectangle can be separated from the pentagon. Then
the remaining pentagon can be changed to a rectangle which
contradict the maximality of the number of rectangles in $D$.
\end{proof}

One can easily find the following inequality.
$$
5(F_5+F_4) \le 4F_8+ 4F_9+5F_{10}+5F_{11}+ \ldots + \lfloor
\frac{n}{2} \rfloor F_n + \ldots .
$$

\section{Link invariants}
\label{invariants}


When links are decorated by the fundamental representations
$V_{\lambda_i}$ of the quantum $\mathfrak{sl}(n)$, denoted by $i$,
Murakami, Ohtsuki and Yamada~\cite{MOY:Homfly} found a quantum
invariant for framed links by resolving each crossing in a link
diagram $D$ of $L$ as shown in Figures~\ref{pstvresol1} and
\ref{pstvresol2}.



For negative crossings, we replace $q$ with $q^{-1}$. $P_n(q)$ is
the special case of $q^{-\omega(D)\frac{n}{2}}[D]_n$, when all
components are colored by the fundamental representation
$V_{\lambda_1}$. They showed that $P_n(q)$ is an isotopy invariant
and that $[D]_n$ is a regular isotopy invariant for other colorings
\cite{MOY:Homfly}. However, one can make it a link invariant by
using a suitable writhe but we have to be careful since there are
more than one colors. For a coloring $\mu$ of a diagram $D$ of a
link $L$, we first consider a \emph{colored writhe} $\omega_i(D)$ as
the sum of writhes of components colored by $i$. Then we set
$$K_n(L,\mu) = \prod_i q^{-\omega_i(D)\frac{i(n-i+1)}{2}} [D]_n,$$
where the product runs over all colors $i$.

\begin{theorem} $K_n(L,\mu)$ is an isotopy invariant of a link $L$, $i. e$, invariant under
Reidermeister moves $I, II$ and  $III$.
\label{invariant}\end{theorem}


\begin{figure}
\begin{eqnarray}\nonumber\left[
\pspicture[.45](-1,-1)(1,1) \rput(-.1,.1){\rnode{a2}{$$}}
\rput(.1,-.1){\rnode{a4}{$$}} \rput(.7,.7){\rnode{b1}{$$}}
\rput(-.7,.7){\rnode{b2}{$$}} \rput(-.7,-.7){\rnode{b3}{$$}}
\rput(.7,-.7){\rnode{b4}{$$}} \rput(.85,.8){\rnode{c1}{$j$}}
\rput(-.8,.8){\rnode{c2}{$i$}} \rput(-.85,-.8){\rnode{c3}{$j$}}
\rput(.8,-.8){\rnode{c4}{$i$}} \ncline{b3}{b1}
\psline[arrowscale=1.5]{->}(.5,.5)(.7,.7) \ncline{b4}{a4}
\psline[arrowscale=1.5]{->}(-.5,.5)(-.7,.7) \ncline{a2}{b2}
\endpspicture\right]_n
&=&\sum_{k=0}^{i} (-1)^{k+(j+1)i}q^{\frac{(i-k)}{2}} \left[
\pspicture[.45](-2,-1.2)(1.8,1.2)\rput(.5,.5){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.5,.5){\rnode{a2}{$$}}
\rput(-.5,-.5){\rnode{a3}{$$}} \rput(.5,-.5){\rnode{a4}{$$}}
\rput(1,1){\rnode{b1}{$$}} \rput(-1,1){\rnode{b2}{$$}}
\rput(-1,-1){\rnode{b3}{$$}} \rput(1,-1){\rnode{b4}{$$}}
\rput(1.15,1.1){\rnode{c1}{$j$}} \rput(-1.1,1.1){\rnode{c2}{$i$}}
\rput(-1.15,-1.1){\rnode{c3}{$j$}} \rput(1.1,-1.1){\rnode{c4}{$i$}}
\rput[a0](0pt,29pt){$j+k-i$} \rput[a0](-33pt,0pt){$j+k$}
\rput[a0](0pt,-25pt){$k$} \rput[a0](33pt,0pt){$i-k$}
\ncline{a1}{b1}\middlearrow \ncline{a2}{b2}\middlearrow
\ncline{b3}{a3}\middlearrow \ncline{b4}{a4}\middlearrow
\ncline{a2}{a1}\middlearrow \ncline{a4}{a1}\middlearrow
\ncline{a3}{a2}\middlearrow \ncline{a4}{a3}\middlearrow
\endpspicture\right]_n
\end{eqnarray}
\caption{Skein expansion case 1 : $j\ge i$.}  \label{pstvresol1}
\end{figure}


\begin{figure}
\begin{eqnarray}\nonumber
\left[ \pspicture[.45](-1,-1)(1,1) \rput(-.1,.1){\rnode{a2}{$$}}
\rput(.1,-.1){\rnode{a4}{$$}}  \rput(.7,.7){\rnode{b1}{$$}}
\rput(-.7,.7){\rnode{b2}{$$}} \rput(-.7,-.7){\rnode{b3}{$$}}
\rput(.7,-.7){\rnode{b4}{$$}} \rput(.85,.8){\rnode{c1}{$j$}}
\rput(-.8,.8){\rnode{c2}{$i$}} \rput(-.85,-.8){\rnode{c3}{$j$}}
\rput(.8,-.8){\rnode{c4}{$i$}}\ncline{b3}{b1}
\psline[arrowscale=1.5]{->}(.5,.5)(.7,.7) \ncline{b4}{a4}
\psline[arrowscale=1.5]{->}(-.5,.5)(-.7,.7) \ncline{a2}{b2}
\endpspicture\right]_n
&=&\sum_{k=0}^{j} (-1)^{k+(i+1)j}q^{\frac{(j-k)}{2}}\left[
\pspicture[.45](-2,-1.2)(1.8,1.2)\rput(.5,.5){\rnode{a1}{$$}}
\rput(0,0){\rnode{a0}{$$}} \rput(-.5,.5){\rnode{a2}{$$}}
\rput(-.5,-.5){\rnode{a3}{$$}} \rput(.5,-.5){\rnode{a4}{$$}}
\rput(1,1){\rnode{b1}{$$}} \rput(-1,1){\rnode{b2}{$$}}
\rput(-1,-1){\rnode{b3}{$$}} \rput(1,-1){\rnode{b4}{$$}}
\rput(1.15,1.1){\rnode{c1}{$j$}} \rput(-1.1,1.1){\rnode{c2}{$i$}}
\rput(-1.15,-1.1){\rnode{c3}{$j$}} \rput(1.1,-1.1){\rnode{c4}{$i$}}
\rput[a0](0pt,29pt){$j+k-i$} \rput[a0](-33pt,0pt){$j-k$}
\rput[a0](0pt,-25pt){$k$} \rput[a0](33pt,0pt){$i+k$}
\ncline{a1}{b1}\middlearrow \ncline{a2}{b2}\middlearrow
\ncline{b3}{a3}\middlearrow \ncline{b4}{a4}\middlearrow
\ncline{a1}{a2}\middlearrow \ncline{a4}{a1}\middlearrow
\ncline{a3}{a2}\middlearrow \ncline{a3}{a4}\middlearrow
\endpspicture\right]_n
\end{eqnarray}
\caption{Skein expansion case 2 : $j< i$.} \label{pstvresol2}
\end{figure}

\begin{figure}
$$
\left[ \pspicture[.45](-.9,-1)(.8,1) \rput(-.1,.1){\rnode{a2}{$$}}
\rput(.1,-.1){\rnode{a4}{$$}}  \rput(.5,.5){\rnode{b1}{$$}}
\rput(-.5,.5){\rnode{b2}{$$}} \rput(-.5,-.5){\rnode{b3}{$$}}
\rput(.5,-.5){\rnode{b4}{$$}} \rput(-.6,.6){\rnode{c2}{$i$}}
\ncline{b3}{b1} \psline[arrowscale=1.5]{->}(.2,.2)(.4,.4)
\ncline{b4}{a4} \psline[arrowscale=1.5]{->}(-.2,.2)(-.4,.4)
\ncline{a2}{b2} \nccurve[angleA=45,angleB=-45]{b1}{b4}
\endpspicture\right]_n = q^{\frac{i(n-i+1)}{2}} \left[ \pspicture[.45](-.9,-1)(-.3,1) \rput(-.1,.1){\rnode{a2}{$$}}
\rput(.1,-.1){\rnode{a4}{$$}}  \rput(.5,.5){\rnode{b1}{$$}}
\rput(-.5,.5){\rnode{b2}{$$}} \rput(-.5,-.5){\rnode{b3}{$$}}
\rput(.5,-.5){\rnode{b4}{$$}} \rput(-.6,.6){\rnode{c2}{$i$}}
\nccurve[angleA=60,angleB=-60]{b3}{b2}\middlearrow
\endpspicture\right]_n \hskip .1cm, \hskip .1cm
\left[ \pspicture[.45](-.9,-1)(.8,1) \rput(.1,.1){\rnode{a1}{$$}}
\rput(-.1,-.1){\rnode{a3}{$$}}  \rput(.5,.5){\rnode{b1}{$$}}
\rput(-.5,.5){\rnode{b2}{$$}} \rput(-.5,-.5){\rnode{b3}{$$}}
\rput(.5,-.5){\rnode{b4}{$$}} \rput(-.6,.6){\rnode{c2}{$i$}}
\ncline{b2}{b4} \psline[arrowscale=1.5]{->}(.2,.2)(.4,.4)
\ncline{b1}{a1} \psline[arrowscale=1.5]{->}(-.2,.2)(-.4,.4)
\ncline{b3}{a3} \nccurve[angleA=45,angleB=-45]{b1}{b4}
\endpspicture\right]_n = q^{-\frac{i(n-i+1)}{2}} \left[ \pspicture[.45](-.9,-1)(-.3,1) \rput(-.1,.1){\rnode{a2}{$$}}
\rput(.1,-.1){\rnode{a4}{$$}}  \rput(.5,.5){\rnode{b1}{$$}}
\rput(-.5,.5){\rnode{b2}{$$}} \rput(-.5,-.5){\rnode{b3}{$$}}
\rput(.5,-.5){\rnode{b4}{$$}} \rput(-.6,.6){\rnode{c2}{$i$}}
\nccurve[angleA=60,angleB=-60]{b3}{b2}\middlearrow
\endpspicture\right]_n
$$
\caption{Resolving Reidermesiter move I.} \label{reider1}
\end{figure}


\begin{proof}
By Theorem~\ref{onedim}, we know $K_n(L,\mu)$ is well defined. It
was shown that $[D]_n$ is a regular isotopy invariant
\cite{MOY:Homfly}, so is $K_n(L,\mu)$. One can find the equations in
Figure~\ref{reider1} by relation~\ref{anbigon1} and a routine
induction. Then $K_n(L,\mu)$ is invariant under Reidermeister move
I.
\end{proof}

\subsection{Proof of Theorem \ref{mainthm}}
Let $\mu$ be a $p$-periodic coloring of $L$. Let $R$ be the
fundamental region by the action $h$. Let $\mathcal{C}$ be the set
of all crossings of $L$ and let $\overline{\mathcal{C}}$ be the set
of all crossings in the region $R$. Let $i(c)$ and $j(c)$ be the
weights of two components at the crossing $c$ as in Figures
\ref{pstvresol1} and \ref{pstvresol2}. Let $J(c)$ be the minimum of
$i(c)$ and $j(c)$ for the crossing $c$. Let $D$ be the diagram after
the expansion by the equations in Figures \ref{pstvresol1} and
\ref{pstvresol2}. Let $D'$ be the diagram obtained by identical
expanding of the crossings which are the same by the action $h$ and
$D''$ be the diagram obtained from $D'$ by identical applications of
relations at the faces which are in the same orbit by the action
$h$. If any expansion of crossings occurs in $R$, it must be used
identically for all other $p-1$ copies of $R$. Otherwise there will
be $p$ identical shapes by the rotation of order $p$, then it is
congruent to zero modulo $p$. This implies the first congruence in
equation \ref{eq1}. By the same philosophy, if any expansion of
relations occurs in $R$, it must be used identically for all other
$p-1$ copies of $R$. Otherwise it is congruent to zero modulo $p$
and this implies the second congruence. Let us remark that we have
not used relations that might occur in the faces which are not
entirely contained in $R$. If there is an unknot in $\overline{D''}$
which was in $D''$, there are $p$ copies in $D''$. Thus raising
$p$-power gives us the equality. But if there is an unknot in
$\overline{D''}$ which was not in $D''$, it should be counted once
in $D''$ and $\overline{D''}$, thus we have to use $\mathcal{I}_n$
congruence.

\begin{align} \label{eq1}
K_n(L,\mu)&= \prod_{c\in \mathcal{C}}\sum_{k=0}^{J(c)}
(-1)^{k+(i(c)+1)j(c)}q^{\frac{(j(c)-k)}{2}} [D]_n \\
\nonumber &\equiv \prod_{\overline{c}\in
\overline{\mathcal{C}}}\sum_{k=0}^{J(\overline{c})}
((-1)^{k+(i(\overline{c})+1)j(\overline{c})}q^{\frac{(j(\overline{c})-k)}{2}})^p
[D']_n  \hskip 1cm(mod \hskip .3cm p)\\ \nonumber &\equiv
\prod_{\overline{c}\in
\overline{\mathcal{C}}}\sum_{k=0}^{J(\overline{c})}
((-1)^{k+(i(\overline{c})+1)j(\overline{c})}q^{\frac{(j(\overline{c})-k)}{2}})^p
[D'']_n  \hskip 1cm(mod \hskip .3cm p) \\ \nonumber &\equiv
\prod_{\overline{c}\in
\overline{\mathcal{C}}}\sum_{k=0}^{J(\overline{c})}
((-1)^{k+(i(\overline{c})+1)j(\overline{c})}q^{\frac{(j(\overline{c})-k)}{2}})^p
([\overline{D''}]_n )^p \hskip 1cm(mod \hskip .3cm \mathcal{I}_n)
\end{align}
where $\overline{D''}= D''/\mathbb{Z}_p$, $\mathbb{Z}_p$ is
generated by the action $h$, and $\mathcal{I}_n$ is the ideal of
$\mathbb{Z}[q^{\pm \frac 12}]$ generated by $p$ and
$\left[\begin{matrix}n\\
i\end{matrix}\right]^p-\left[\begin{matrix}n\\
i\end{matrix}\right]$ for $i=1, 2, \ldots, \lfloor \frac{n}{2}
\rfloor$. The following congruences should be straightforward.

\begin{align*}
&\equiv  \prod_{\overline{c}\in
\overline{\mathcal{C}}}\sum_{k=0}^{J(\overline{c})}
((-1)^{k+(i(\overline{c})+1)j(\overline{c})}q^{\frac{(j(\overline{c})-k)}{2}} [\overline{D}]_n )^p \hskip 1cm(mod \hskip .3cm p)\\
&\equiv  (\prod_{\overline{c}\in
\overline{\mathcal{C}}}\sum_{k=0}^{J(\overline{c})}
(-1)^{k+(i(\overline{c})+1)j(\overline{c})}q^{\frac{(j(\overline{c})-k)}{2}} [\overline{D}]_n )^p \hskip 1cm(mod \hskip .3cm p)\\
&\equiv  (K_n(\overline{L},\overline{\mu}))^p  \hskip 1cm(mod
\hskip .3cm p)\end{align*}


Next, we will extend Theorem \ref{mainthm} to the colored
$\mathfrak{sl}(n)$ HOMFLY polynomial of a periodic link whose
components are colored by irreducible representations of quantum
$\mathfrak{sl}(n)$. For a given colored link $L$ of $l$ components
say, $L_1, L_2, \ldots, L_l$, where each component $L_i$ is colored
by an irreducible representation $V_{a(i)_1\lambda_1 +
a(i)_2\lambda_2 + \ldots + a(i)_{n-1}\lambda_{n-1}}$ of
$\mathfrak{sl}(n)$ and $\lambda_1, \lambda_2, \ldots, \lambda_{n-1}$
are the fundamental weights of $\mathfrak{sl}(n)$. The coloring is
denoted by $\mu=(a(1)_1\lambda_1 + a(1)_2\lambda_2 + \ldots +
a(1)_{n-1}\lambda_{n-1}, a(2)_1\lambda_1 + a(2)_2\lambda_2 + \ldots
+ a(2)_{n-1}\lambda_{n-1}, \ldots, a(l)_1\lambda_1 + a(l)_2\lambda_2
+ \ldots + a(l)_{n-1}\lambda_{n-1})$. First we replace each
component $L_i$ by $a(i)_1+a(i)_2+\ldots +a(i)_{n-1}$ copies of
parallel lines and each $a(i)_j$ line is colored by the weight
$\lambda_j$. Then we put a clasp of weight $(a(i)_1\lambda_1 +
a(i)_2\lambda_2 + \ldots + a(i)_{n-1}\lambda_{n-1})$ for $L_i$. If
we assume the clasps are far away from crossings, we expand each
crossing as in Figure~\ref{pstvresol1} and \ref{pstvresol2}, then
clasps \cite{yokota:skeinforn}. The value we find after removing all
faces by using the relations is \emph{the colored $\mathfrak{sl}(n)$
HOMFLY polynomial} $G_n(L,\mu)$ of $L$.

\begin{theorem}
Let $p$ be a positive integer and $L$ be a $p-$periodic link in
$S^3$ with the factor link $\overline{L}$. Let $\mu$ be a
$p$-periodic coloring of $L$ and $\overline{\mu}$ be the induced
coloring of $\overline{L}$. Then for $n\ge 2$,

$$G_n(L,\mu) \cong G_n(L,\overline{\mu})^p \hskip ,7cm modulo \hskip .2cm
\mathcal{I}_n,$$ where $\overline L$ is the factor link and
$\mathcal{I}_n$ is the ideal of $\mathbb{Z}[q^{\pm \frac 12}]$
generated by $p$ and
$\left[\begin{matrix}n\\
i\end{matrix}\right]^p-\left[\begin{matrix}n\\
i\end{matrix}\right]$ for $i=1, 2, \ldots, \lfloor \frac{n}{2}
\rfloor$. \label{main2}
\end{theorem}
\begin{proof}
Since all clasps are idempotents, we put $p$-copies of clasps to
each copy of the fundamental region by the action $h$ of the
periodicity of $L$. Thus without expanding the clasps, we obtain the
theorem by the same idea of the proof of the Theorem~\ref{mainthm}.
\end{proof}


\subsection{Proof of Theorem \ref{mirror2}}

For a colored link diagram $D$ we denote its mirror image by $D^*$
which is the colored link diagram obtained from $D$ by changing all
of the crossings. We study another necessary condition for a colored
link to be periodic by using the invariant $K_n(L,q)$. Let $L$ be a
colored link diagram with a crossing $x$. Let $L_+,~L_-,~$ and $L_0$
be the link diagrams obtained by resolving the crossing $x$ as shown
in Figure \ref{firstrel} respectively. Then from the Figure
\ref{pstvresol1}, \ref{pstvresol2}, we see that there exist web
diagrams $L_1,\cdots,L_m$ and polynomials $f_1,\cdots,f_m \in
Z[q^{\pm \frac{1}{2}}]$ for some positive integer $m$ such that
$$[L_+]_n - [L_-]_n =
(q^{\frac{1}{2}}-q^{-\frac{1}{2}})(f_1[L_1]_n+\cdots+f_m[L_m]_n).$$


In particular, by applying this relation repeatedly for a colored
periodic link we obtain the following lemma.

\begin{lemma} \label{mirror1}
For a colored $p$--periodic link diagram $D$ and its mirror image
$D^*$, we have $$[D]_n \equiv [D^*]_n~~\mathrm{mod}~~ (p, q^p-1).$$
\end{lemma}
\begin{proof}
Let $\pi$ be the quotient map of $h$ and $\overline{D}$ be the
factor link of $D$. Then $\pi^{-1}(\overline{D}) = D$. We fix a
crossing $x$ for a nontrivial diagram $\overline{D}$, and let
$\overline{D}_+$ and $\overline{D}_-$ be the colored link diagram
obtained by changing the crossing $x$ to a positive crossing and a
negative crossing respectively.

Let $\overline{D}_1,\cdots,\overline{D}_m$ be web diagrams and let
$f_1,\cdots,f_m \in Z[q^{\pm \frac{1}{2}}]$ be polynomials for some
positive integer $m$ such that
$$[\overline{D}_+]_n - [\overline{D}_-]_n =
(q^{\frac{1}{2}}-q^{-\frac{1}{2}})(f_1[\overline{D}_1]_n+\cdots+f_m[\overline{D}_m]_n).$$

Then by considering the periodic action induced from the map $h$, we
find the following congruence relation.

\begin{eqnarray*}
& &[\pi^{-1}(\overline{D}_+)]_n - [\pi^{-1}(\overline{D}_-)]_n \\
 &\equiv&
(q^{\frac{1}{2}}-q^{-\frac{1}{2}})^p(f_1^p[\pi^{-1}(\overline{D}_1)]_n
+\cdots+f_m^p[\pi^{-1}(\overline{D}_m)]_n)~~\mathrm{mod}~~ (p)\\
&\equiv& 0 ~~\mathrm{mod}~~ (p, q^p-1).
\end{eqnarray*}

Since $[\pi^{-1}(\overline{D}_+)]_n \equiv
[\pi^{-1}(\overline{D}_-)]_n ~~\mathrm{mod}~~ (p, q^p-1)$ for each
crossing of $\overline{D}$ and since there are finitely many
crossings in $\pi^{-1}(\overline{D})$, we see that
$[\pi^{-1}(\overline{D})]_n \equiv [\pi^{-1}(\overline{D}^*)]_n
~~\mathrm{mod}~~ (p, q^p-1)$, where $\overline{D}^*$ is the mirror
image of $\overline{D}$. Therefore we have $$[D]_n \equiv [D^*]_n
~~\mathrm{mod}~~ (p, q^p-1).$$
\end{proof}


Now, we are ready to prove Theorem \ref{main2}. For a tangle $T$, we
denote its closure by $CL(T)$ if it is well defined. Let $L$ be a
$p$--periodic link and $T$ be a tangle such that $L$ is the closure
$Cl(T^p)$ of the tangle $T^p$ which is $p$ times self-product of
$T$. Let $D$ be a diagram of $L$ and $x$ be a crossing of the
diagram $D$. Let $T_+$ and $T_-$ be the diagram obtained from the
diagram of $T$ by changing the crossing $x$ to a positive crossing
and negative crossing respectively. If the two colorings of the
strands near the crossing $x$ are different then by using Lemma
\ref{mirror1}, we see that

\begin{eqnarray*}
& &K_n(Cl({(T_+)^p}), \mu) - K_n(Cl({(T_-)^p}), \mu) \\
&=&\Pi_i
q^{-w_i(D)\frac{i(n-i+1)}{2}}([Cl({(T_+)^p})]_n-[Cl({(T_-)^p})]_n)\\
&\equiv&0 ~~\mathrm{mod}~~ (p, q^p-1).
\end{eqnarray*}

If two colorings of the strands near the crossing $x$ are equal, say
$i$, then $w_i(Cl({(T_+)^p})) = w_i(Cl({(T_+)^p})) +2p$. Then we see
that there exist a polynomial $g\in \mathbb{Z}[q^{\pm \frac{1}{2}}]$
such that

\begin{eqnarray*}
& & K_n(Cl({(T_+)^p}), \mu) - K_n(Cl({(T_-)^p}), \mu)\\ &=&
g([Cl({(T_+)^p})]_n-q^{pi(n-i+1)}[Cl({(T_-)^p})]_n)\\
&\equiv& 0 ~~\mathrm{mod}~~ (p, q^p-1).
\end{eqnarray*}

The last congruence relation in the above formulae can be obtained
by using Lemma \ref{mirror1}. Thus we see that

$$K_n(Cl({(T_+)^p}), \mu) \equiv K_n(Cl({(T_-)^p}), \mu)~~\mathrm{mod}~~ (p, q^p-1).$$

Since $T$ can be obtained from its mirror image $T^*$ by a finite
sequence of crossing changes, we get
$$K_n(L,\mu) \equiv K_n(L^*,\mu^*)  ~~\mathrm{mod}~~ (p, q^p-1).$$

It completes the proof the theorem.

\vskip 1cm
 \noindent{\bf Acknowledgements}

The authors would like to thank Greg Kuperberg for introducing the
subject and helpful discussion, Mikhail Khovanov, Jaeun Lee, Hitoshi
Murakami, Myungsoo Seo for their attention to this work. Also, the
\TeX\, macro package PSTricks~\cite{PSTricks} was essential for
typesetting the equations and figures.

\bibliographystyle{amsplain}
\begin{thebibliography}{100}
\bibitem{chbili} N. Chbili, \textit{The quantum SU(3) invariant of links and Murasugi's congruence},
Topology appl., 122, (2002), 479--485.

\bibitem{CL:period} Q. Chen and T. Le, \textit{Quantum invariants and periodic links and periodic manifolds}, preprint,
arXiv:math.QA/0408358.

\bibitem{khvanov:canonical} I. Frenkel and M. Khovanov, \textit{Canonical bases in tensor products and
graphical calculus for $U_{q}(\mathfrak{sl}_2)$}, Duke Math. J.,
87(3), (1997) 409--480.

\bibitem{FultonHarris:gtm} W. Fulton and J. Harris, \textit{ Representation theory},
Graduate Texts in Mathematics, 129, Springer-Verlag, New
York-Heidelberg-Berlin, 1991.

\bibitem{Jones:braid} V. F. R. Jones, \textit{Hecke algebra representations of braid groups and link polynomials},
Ann. of Math., 126, (1987), 335--388.

\bibitem{JP5} M.--J. Jeong and C.--Y. Park, \textit{Lens knots,
periodic knots and Vassiliev invariants}, J. of Knot Theory and Its
Ramifications, Vol. 13 (2004), 1041--1056.

\bibitem{khvanov:comm} M. Khovanov, private communication.

\bibitem{Dongseok:thesis} D. Kim, \textit{Graphical Calculus on Representations of Quantum Lie
Algebras}, Thesis, UCDavis, 2003, arXiv:math.QA/0310143.

\bibitem{Kuperberg:spiders} G. Kuperberg, \textit{Spiders for rank 2 {Lie} algebras}, Comm. Math. Phys.,
180(1), (1996) 109--151, arXiv:q-alg/9712003.

\bibitem{KRT:knot} C. Kassel, M. Rosso and V. Turaev, \textit{Quantum groups and knot invariants},
Panoramas et Syntheses, 5, Societe Mathematique de France, 1997.

\bibitem{KR:factor} M. Khovanov and L. Rozansky, \textit{Matrix factorizations and link homology},
preprint, arXiv:QA/0401268.

\bibitem{mu:alexander} K. Murasugi, \textit{On periodic knots}, Comment. Math. Helv.,
46 (1971), 162--174.

\bibitem{mu:jones} K. Murasugi, \textit{The Jones polynomials of periodic links},
Pacific J. Math., 131 (1988), 319--329.

\bibitem{MOY:Homfly} H. Murakami and T. Ohtsuki and S. Yamada, \textit{HOMFLY polynomial via an invariant of colored plane graphs},
L'Enseignement Mathematique, t., 44 (1998), 325--360.

\bibitem{Przytycki:criterion} J. H. Przytycki, \textit{On Murasugi's and Traczyk's criteria for
periodic links}, Math. Ann. 283 (1989), 465--478.

\bibitem{PS:sun} J. H. Przytycki and A. Sikora, \textit{$SU_n$-quantum invariants for
periodic links}, Preprint.

\bibitem{PSTricks}
T.~{Van Zandt}. PSTricks: {PostScript} macros for generic {\TeX}.
Available at {\tt ftp ://ftp.princeton.edu/pub/tvz/}.

\bibitem{RT:1} N. Yu. Reshetikhin and V. G. Turaev,
\textit{Ribbob graphs and their invariants derived from quantum
groups}, Comm. Math. Phys., 127 (1990), 1--26.

\bibitem{RT:2} N. Yu. Reshetikhin and V. G. Turaev,
\textit{Invariants of $3$-manifolds via link polynomials and quantum
groups}, Invent. Math., 103 (1991), 547--597.

\bibitem{Sikora:sln} A. Sikora.,
\textit{Skein theory for $SU(n)$-quantum invariants}, Algebr. Geom.
Topol. 5 (2005) 865--897, arXiv:math.QA/0407299.

\bibitem{traczyk:period3} P. Traczyk, \textit{A criterion for knots of period $3$}, Topology and its
Appl. 36 (1990), 275--281.

\bibitem{yokota:skein} Y. Yokota, \textit{The skein polynomial of periodic knots}, Math. Ann.
291(2) (1991), 281--291.

\bibitem{yokota:kauffman} Y. Yokota, \textit{The Kauffman polynomial of periodic knots}, Topology
32(2) (1993), 309--324.

\bibitem{yokota:poly} Y. Yokota, \textit{Polynomila invariants of periodic knots}, J. Knot
Theory and Its Ramifications 5(4) (1996), 553--567.

\bibitem{yokota:skeinforn} Y. Yokota, \textit{Skein and quantum
$SU(N)$ invariants of 3-manifolds}, Math. Ann., 307 (1997),
109--138.



\end{thebibliography}

\end{document}
%starshea@tky3.3web.ne.jp <starshea@tky3.3web.ne.jp>

math0506403.extracted_bib
\begin{thebibliography}{100}
\bibitem{chbili} N. Chbili, \textit{The quantum SU(3) invariant of links and Murasugi's congruence},
Topology appl., 122, (2002), 479--485.

\bibitem{CL:period} Q. Chen and T. Le, \textit{Quantum invariants and periodic links and periodic manifolds}, preprint,
arXiv:math.QA/0408358.

\bibitem{khvanov:canonical} I. Frenkel and M. Khovanov, \textit{Canonical bases in tensor products and
graphical calculus for $U_{q}(\mathfrak{sl}_2)$}, Duke Math. J.,
87(3), (1997) 409--480.

\bibitem{FultonHarris:gtm} W. Fulton and J. Harris, \textit{ Representation theory},
Graduate Texts in Mathematics, 129, Springer-Verlag, New
York-Heidelberg-Berlin, 1991.

\bibitem{Jones:braid} V. F. R. Jones, \textit{Hecke algebra representations of braid groups and link polynomials},
Ann. of Math., 126, (1987), 335--388.

\bibitem{JP5} M.--J. Jeong and C.--Y. Park, \textit{Lens knots,
periodic knots and Vassiliev invariants}, J. of Knot Theory and Its
Ramifications, Vol. 13 (2004), 1041--1056.

\bibitem{khvanov:comm} M. Khovanov, private communication.

\bibitem{Dongseok:thesis} D. Kim, \textit{Graphical Calculus on Representations of Quantum Lie
Algebras}, Thesis, UCDavis, 2003, arXiv:math.QA/0310143.

\bibitem{Kuperberg:spiders} G. Kuperberg, \textit{Spiders for rank 2 {Lie} algebras}, Comm. Math. Phys.,
180(1), (1996) 109--151, arXiv:q-alg/9712003.

\bibitem{KRT:knot} C. Kassel, M. Rosso and V. Turaev, \textit{Quantum groups and knot invariants},
Panoramas et Syntheses, 5, Societe Mathematique de France, 1997.

\bibitem{KR:factor} M. Khovanov and L. Rozansky, \textit{Matrix factorizations and link homology},
preprint, arXiv:QA/0401268.

\bibitem{mu:alexander} K. Murasugi, \textit{On periodic knots}, Comment. Math. Helv.,
46 (1971), 162--174.

\bibitem{mu:jones} K. Murasugi, \textit{The Jones polynomials of periodic links},
Pacific J. Math., 131 (1988), 319--329.

\bibitem{MOY:Homfly} H. Murakami and T. Ohtsuki and S. Yamada, \textit{HOMFLY polynomial via an invariant of colored plane graphs},
L'Enseignement Mathematique, t., 44 (1998), 325--360.

\bibitem{Przytycki:criterion} J. H. Przytycki, \textit{On Murasugi's and Traczyk's criteria for
periodic links}, Math. Ann. 283 (1989), 465--478.

\bibitem{PS:sun} J. H. Przytycki and A. Sikora, \textit{$SU_n$-quantum invariants for
periodic links}, Preprint.

\bibitem{PSTricks}
T.~{Van Zandt}. PSTricks: {PostScript} macros for generic {\TeX}.
Available at {\tt ftp ://ftp.princeton.edu/pub/tvz/}.

\bibitem{RT:1} N. Yu. Reshetikhin and V. G. Turaev,
\textit{Ribbob graphs and their invariants derived from quantum
groups}, Comm. Math. Phys., 127 (1990), 1--26.

\bibitem{RT:2} N. Yu. Reshetikhin and V. G. Turaev,
\textit{Invariants of $3$-manifolds via link polynomials and quantum
groups}, Invent. Math., 103 (1991), 547--597.

\bibitem{Sikora:sln} A. Sikora.,
\textit{Skein theory for $SU(n)$-quantum invariants}, Algebr. Geom.
Topol. 5 (2005) 865--897, arXiv:math.QA/0407299.

\bibitem{traczyk:period3} P. Traczyk, \textit{A criterion for knots of period $3$}, Topology and its
Appl. 36 (1990), 275--281.

\bibitem{yokota:skein} Y. Yokota, \textit{The skein polynomial of periodic knots}, Math. Ann.
291(2) (1991), 281--291.

\bibitem{yokota:kauffman} Y. Yokota, \textit{The Kauffman polynomial of periodic knots}, Topology
32(2) (1993), 309--324.

\bibitem{yokota:poly} Y. Yokota, \textit{Polynomila invariants of periodic knots}, J. Knot
Theory and Its Ramifications 5(4) (1996), 553--567.

\bibitem{yokota:skeinforn} Y. Yokota, \textit{Skein and quantum
$SU(N)$ invariants of 3-manifolds}, Math. Ann., 307 (1997),
109--138.



\end{thebibliography}