The $\mathfrak{gl}_2$-Skein Module of Lens Spaces via the Torus and Solid Torus

TL;DR

This paper calculates the $rak{gl}_2$-skein module of lens spaces by analyzing the algebraic action on the solid torus, revealing a specific spanning set size for these modules.

Contribution

It provides an explicit computation of the $rak{gl}_2$-skein modules of lens spaces using the torus and solid torus frameworks, detailing their structure and basis.

Findings

The $rak{gl}_2$-skein module of lens spaces is spanned by a finite set of elements.

The size of the spanning set depends on the parameter p, specifically $igl(ig floor{rac{p}{2}}igl+1igr)igl(2ig floor{rac{p}{2}}igl+1igr)$.

The action of the skein algebra of the torus on the solid torus module is explicitly computed.

Abstract

We compute the action of the $\mathfrak{gl}_2$-skein algebra of the torus on the $\mathfrak{gl}_2$-skein module of the solid torus. As a result, we show that the $\mathfrak{gl}_2$-skein modules of lens spaces is spanned by $\left(\left\lfloor{\frac{p}{2}}\right \rfloor+1\right)\left(2\left\lfloor{\frac{p}{2}}\right \rfloor+1\right)$ elements.