The Kauffman Bracket Skein Module of S1 × S2 via Braids

Abstract

In this paper, we present two different ways for computing the Kauffman bracket skein module of S1×S2, KBSMS1×S2, via braids. We first extend the universal Kauffman bracket type invariant V for knots and links in the Solid Torus ST, which is obtained via a unique Markov trace constructed on the generalized Temperley–Lieb algebra of type B, to an invariant for knots and links in S1×S2. We do that by imposing on V relations coming from the braid band moves. These moves reflect isotopy in S1×S2 and they are similar to the second Kirby move. We obtain an infinite system of equations, a solution of which is equivalent to computing KBSMS1×S2. We show that KBSMS1×S2 is not torsion free and that its free part is generated by the unknot (or the empty knot). We then present a diagrammatic method for computing KBSMS1×S2 via braids. Using this diagrammatic method, we also obtain a closed formula for the torsion part of KBSMS1×S2.