An alternative basis for the Kauffman bracket skein module of the Solid Torus via braids

TL;DR

This paper introduces a new basis for the Kauffman bracket skein module of the solid torus, utilizing the Temperley-Lieb algebra of type B, to facilitate computations for lens spaces via braids.

Contribution

It presents an alternative basis for the skein module of the solid torus using type B Temperley-Lieb algebra, aiding in calculations for lens spaces.

Findings

New basis for Kauffman bracket skein module of solid torus

Utilizes Temperley-Lieb algebra of type B

Simplifies computations for lens spaces

Abstract

In this paper we give an alternative basis, $\mathcal{B}_{\rm ST}$, for the Kauffman bracket skein module of the solid torus, ${\rm KBSM}\left({\rm ST}\right)$. The basis $\mathcal{B}_{\rm ST}$ is obtained with the use of the Tempereley--Lieb algebra of type B and it is appropriate for computing the Kauffman bracket skein module of the lens spaces $L(p, q)$ via braids.