KBSM of lens spaces $L(p,2)$ and $L(4k,2k+1)$

TL;DR

This paper constructs new bases for the Kauffman bracket skein modules of specific lens spaces, providing explicit decompositions and advancing the understanding of their algebraic structures.

Contribution

It introduces new bases for the KBSM of lens spaces $L(p,2)$ and $L(4k,2k+1)$, and a new generating set for $S^2 \times S^1$, enhancing prior computations.

Findings

New bases for KBSM of $L(p,2)$ and $L(4k,2k+1)$ lens spaces.

Decomposition of KBSM of $S^2 \times S^1$ into cyclic modules.

Explicit algebraic structures for these skein modules.

Abstract

J. Hoste and J. H. Przytycki computed the Kauffman bracket skein module (KBSM) of lens spaces in their papers published in 1993 and 1995. Using a basis for the KBSM of a fibered torus, we construct new bases for the KBSMs of two families of lens spaces: $L(p,2)$ and $L(4k,2k+1)$ with $k\neq 0$. For KBSM of $L(0,1) = {\bf S}^{2}\times S^{1}$, we find a new generating set that yields its decomposition into a direct sum of cyclic modules.