Basis for KBSM of fibered torus with multiplicity two exceptional fiber

TL;DR

This work targets the Kauffman bracket skein module (KBSM) of a $(\beta,2)$-fibered torus by developing a combinatorial arrow-diagram framework. It first builds a one-parameter family of bases for the KBSM of $\mathbf{A}^{2}\times S^{1}$, using a modified bracket and explicit polynomials to express arrow diagrams; each basis $\Sigma_{c}$ yields a concrete isomorphism $\mathcal{S}_{2,\infty}(\mathbf{A}^{2}\times S^{1};R,A) \cong R\Sigma_{c}$. The method is then extended to the $(\beta,2)$-fibered torus $V(\beta,2)$ by introducing a marked-disk presentation with $S_{\beta}$-moves and demonstrating that $\mathcal{S}_{2,\infty}(V(\beta,2);R,A) \cong R\Sigma'_{\nu}$ with $\nu=\lfloor\beta/2\rfloor$, via a map $\phi_{\beta}$ that respects skein and isotopy relations. Together, these results provide explicit, computable bases for KBSM on fibered tori, paving the way to compute KBSM for lens spaces and small Seifert fibered 3-manifolds in subsequent work.

Abstract

We construct a family of bases for the Kauffman bracket skein module (KBSM) of the product of an annulus and a circle. Using these bases, we find a new basis for the KBSM of $(β,2)$-fibered torus as a first step toward developing techniques for computing KBSM of a family of small Seifert fibered $3$-manifolds.

Figures (37)

• Figure 1.1: Skein triple $L_{+}$, $L_{0}$, $L_{\infty}$ and relation $L\sqcup T_{1} + (A^{-2}+A^{2})L$
• Figure 2.1: Arrow diagram of a link in ${\bf A}^{2}\times S^{1}$
• Figure 2.2: Arrow moves $\Omega_{1}-\Omega_{5}$
• Figure 2.3: Simplified arrow diagram
• Figure 2.4: $S_{\beta}$-move on ${\bf D}^{2}_{\beta}$ with $\beta = 5$

Theorems & Definitions (39)

• Lemma 2.1
• Theorem 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Definition 3.3
• Lemma 3.4
• proof
• Lemma 3.5