Basis for KBSM of fibered torus with multiplicity two exceptional fiber

TL;DR

This paper develops new bases for the Kauffman bracket skein module of fibered tori, aiding in the computation of skein modules for small Seifert fibered 3-manifolds, advancing understanding in quantum topology.

Contribution

It introduces a novel basis for the KBSM of fibered tori, facilitating calculations for a class of Seifert fibered 3-manifolds.

Findings

Constructed bases for KBSM of annulus times circle

Derived a new basis for KBSM of (β,2)-fibered torus

Paves the way for computing KBSM of small Seifert fibered manifolds

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 $(\beta,2)$-fibered torus as a first step toward developing techniques for computing KBSM of a family of small Seifert fibered $3$-manifolds.