The Kauffman bracket skein module of $S^1\times S^2$ via braids

TL;DR

This paper develops two braid-based methods to compute the Kauffman bracket skein module of $S^1\times S^2$, revealing its structure and providing explicit formulas for its torsion components.

Contribution

It extends the universal Kauffman bracket invariant to $S^1\times S^2$ using braid moves and introduces a diagrammatic method for computation.

Findings

KBSM of $S^1\times S^2$ is not torsion free.

The free part of KBSM is generated by the unknot.

A closed formula for the torsion part of KBSM is provided.

Abstract

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