The Kauffman bracket skein module of the lens spaces via unoriented braids

Abstract

In this paper we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces $L(p,q)$, KBSM($L(p,q)$), for $q\neq 0$. For doing this, we introduce a new concept, that of an {\it unoriented braid}. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the {\it generalized Temperley-Lieb algebra of type B}, $TL_{1, n}$, which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket type invariant, $V$, for knots and links in ST, via a unique Markov trace constructed on $TL_{1, n}$. The universal invariant $V$ is equivalent to the KBSM(ST). For passing now to the KBSM($L(p,q)$), we impose on $V$ relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in $L(p,q)$ but not in ST, and which reflect the surgery description of $L(p,q)$, obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM($L(p,q)$). We first present the solution for the case $q=1$, which corresponds to obtaining a new basis, $\mathcal{B}_{p}$, for KBSM($L(p,1)$) with $(\lfloor p/2 \rfloor +1)$ elements. We note that the basis $\mathcal{B}_{p}$ is different from the one obtained by Hoste \& Przytycki. For dealing with the complexity of the infinite system for the case $q>1$, we first show how the new basis $\mathcal{B}_{p}$ of KBSM($L(p,1)$) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case $q>1$.