The Kauffman bracket skein module of S1xS2 via braids

In this paper we present two different ways for computing the Kauffman bracket skein module of S1xS2, KBSM(S1xS2), 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 S1xS2. We do that by imposing on V relations coming from the braid band moves. These moves reflect isotopy in S1xS2 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(S1xS2). We show that KBSM(S1xS2) 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(S1xS2) via braids. Using this diagrammatic method we also obtain a closed formula for the torsion part of KBSM(S1xS2).