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

TL;DR

The paper determines the Kauffman bracket skein module $\mathcal{S}_{2,\infty}$ over $\mathbb{Z}[A^{\pm1}]$ for the closed 3-manifold $M = (S^1\times S^2)\#(S^1\times S^2)$. Building from a genus-2 handlebody and two 2-handle attachments along $\beta$ and $\eta$, the authors reduce the computation to explicit handle-sliding relations computed via relative skein modules, employing a pants surface basis and Chebyshev polynomials. They prove that $\mathcal{S}_{2,\infty}(M)$ does not split as a direct sum of a free and torsion submodule, and they exhibit two families of torsion elements, including a new $(1-A^2)$-torsion family. In addition, they provide a counterexample to Marché’s conjecture on the free-plus-torsion decomposition and outline directions for extending the analysis to other connected sums, with potential applications to traces and quantum invariants. The appendix delivers the detailed recurrence-based calculations that yield the explicit expressions for the handle-sliding relations used in the main result.

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.

Figures (20)

• Figure 2.1: Skein triple for the Kauffman bracket skein module.
• Figure 3.1: The 3-manifold $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$ is obtained by gluing a $2$-handle each to $\partial H_2$ along the curves $\beta$ and $\eta$.
• Figure 3.2: .
• Figure 3.3: Marked points $u$ and $v$ on the simple closed curve $\beta$ in $\partial H_2$ that divide it into curves $\beta_1$ and $\beta_2$.
• Figure 3.4: Illustration of $\omega(\alpha)$.

Theorems & Definitions (71)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Example 2.4
• Theorem 2.5
• Definition 2.6
• Remark 2.7
• Theorem 2.8
• Theorem 2.9
• Theorem 3.1