On the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$

TL;DR

This paper computes the Kauffman bracket skein module of a specific non-prime 3-manifold, revealing its complex structure and showing it does not decompose into free and torsion parts, with explicit torsion elements identified.

Contribution

It provides the first detailed computation of the skein module for the connected sum of two S^1×S^2 manifolds, including basis analysis and torsion element identification.

Findings

Computed the skein module of (S^1×S^2)#(S^1×S^2) over Z[A^{±1}]

Showed the skein module does not split into free and torsion parts

Identified two families of torsion elements

Abstract

Determining the structure of the Kauffman bracket skein module of all $3$-manifolds over the ring of Laurent polynomials $\mathbb Z[A^{\pm 1}]$ is a big open problem in skein theory. Very little is known about the skein module of non-prime manifolds over this ring. In this paper, we compute the Kauffman bracket skein module of the $3$-manifold $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$ over the ring $\mathbb Z[A^{\pm 1}]$. We do this by analysing the submodule of handle sliding relations, for which we provide a suitable basis. Along the way we compute the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times D^2)$. We also show that the skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$ does not split into the sum of free and torsion submodules. Furthermore, we illustrate two families of torsion elements in this skein module.