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

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.