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 6 = 0. For doing this, we introduce a new concept, that of an unoriented braid . Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the 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, B p , for KBSM( L ( p, 1)) with ( ⌊ p/ 2 ⌋ +1) elements. We note that the basis 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 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. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements.