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

TL;DR

This paper computes the HOMFLYPT skein module of the 3-manifold S^1 x S^2 using braid techniques, extending invariants from the solid torus and revealing its algebraic structure.

Contribution

It extends the Lambropoulou invariant to S^1 x S^2 and provides a detailed algebraic description of its skein module.

Findings

The free part of the skein module is generated by the empty link.

All other elements in the skein module are torsion.

The computation uses an infinite system of equations related to band moves.

Abstract

In this paper we compute the HOMFLYPT skein module of $S^1 \times S^2\, \cong \, L(0, 1)$, denoted $\mathcal{S}(S^1 \times S^2)$, using braid-theoretic techniques. We extend the Lambropoulou invariant, $X$, for links in the solid torus ST to links in $S^1 \times S^2$, by solving an infinite system of equations of the form $X_{\widehat{a}} = X_{\widehat{bbm(a)}}$, where $bbm(a)$ denotes all possible band moves applied to $a$, for all $a$ in a basis of $\mathcal{S}(ST)$. We show that the free part of $\mathcal{S}(S^1 \times S^2)$ is generated by the empty link, while all other elements are torsion.