Kauffman bracket skein module of the connected sum of two solid tori

TL;DR

This paper determines the structure of the Kauffman bracket skein module for the connected sum of two solid tori, confirming a conjecture and enabling future computations for more complex 3-manifolds.

Contribution

It provides a complete description of the skein module for a specific 3-manifold, confirming a prior conjecture and advancing the understanding of skein modules.

Findings

Structured the skein module of the connected sum of two solid tori.

Proved a conjecture regarding the skein module structure.

Laid groundwork for computing skein modules of arbitrary connected sums.

Abstract

We determine the structure of the Kauffman bracket skein module of the connected sum of two genus one handlebodies over the ring of Laurent polynomials $\mathbb Z[q^{\pm 1}]$, thereby proving a conjecture posed by the first and third authors. Our results lay the groundwork for computing the Kauffman bracket skein module of arbitrary connected sums over the ring $\mathbb Z[q^{\pm 1}]$.