The Kauffman Bracket Skein Module of S1 × S2 via Braids

In this paper, we present two different ways for computing the Kauffman bracket skein module of S^1×S^2, KBSM(S^1×S^2), 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 S^1×S^2. We do that by imposing on V relations coming from the braid band moves. These moves reflect isotopy in S^1×S^2 and they are similar to the second Kirby move. We obtain an infinite system of equations, a solution of which is equivalent to computing KBSM(S^1×S^2). We show that KBSM(S^1×S^2) 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 KBSM(S^1×S^2) via braids. Using this diagrammatic method, we also obtain a closed formula for the torsion part of
KBSM(S^1×S^2).