On torsion in the Kauffman bracket skein module of $3$-manifolds

TL;DR

This paper investigates the conditions under which the Kauffman bracket skein module of 3-manifolds contains torsion, linking it to the size of the $SL_2(\mathbb{C})$-character variety and essential surfaces.

Contribution

It introduces new criteria for torsion in skein modules based on character varieties and essential surfaces, providing counterexamples and revisiting prior work.

Findings

Torsion in skein modules relates to the size of the $SL_2(\mathbb{C})$-character variety.

Manifolds with incompressible tori have new effective torsion criteria.

$\mathcal{S}(\mathbb{RP}^3\# L(p,1))$ has torsion when $p$ is even.

Abstract

We study Kirby problems 1.92(E)-(G), which, roughly speaking, ask for which compact oriented $3$-manifold $M$ the Kauffman bracket skein module $\mathcal{S}(M)$ has torsion as a $\mathbb{Z}[A^{\pm 1}]$-module. We give new criteria for the presence of torsion in terms of how large the $SL_2(\mathbb{C})$-character variety of $M$ is. This gives many counterexamples to question 1.92(G)-(i) in Kirby's list. For manifolds with incompressible tori, we give new effective criteria for the presence of torsion, revisiting the work of Przytycki and Veve. We also show that $\mathcal{S}(\mathbb{R P}^3# L(p,1))$ has torsion when $p$ is even. Finally, we show that for $M$ an oriented Seifert manifold, closed or with boundary, $\mathcal{S}(M)$ has torsion if and only if $M$ admits a $2$-sided non-boundary parallel essential surface.