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 [Formula: see text], KBSM([Formula: see text]), for [Formula: see text]. 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, [Formula: see text], which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, [Formula: see text], for knots and links in ST, via a unique Markov trace constructed on [Formula: see text]. The universal invariant [Formula: see text] is equivalent to the KBSM(ST). For passing now to the KBSM([Formula: see text]), we impose on [Formula: see text] relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in [Formula: see text] but not in ST, and which reflect the surgery description of [Formula: see text], obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM([Formula: see text]). We first present the solution for the case [Formula: see text], which corresponds to obtaining a new basis, [Formula: see text], for KBSM([Formula: see text]) with [Formula: see text] elements. We note that the basis [Formula: see text] is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case [Formula: see text], we first show how the new basis [Formula: see text] of KBSM([Formula: see text]) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case [Formula: see text]. 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.