Skein module dimensions of mapping tori of the 2-torus

Abstract

We determine the dimension of the Kauffman bracket skein module at generic $q$ for mapping tori of the 2-torus, generalizing the well known computation of Carrega and Gilmer. In the process, we give a decomposition of the twisted Hochschild homology of the $G$-skein algebra for $G={\mathrm{SL}}_N$ or ${\mathrm{GL}}_N$, which is a direct summand of the whole skein module, and from which the dimensions follow easily in the cases $G={\mathrm{SL}}_2$ and $G={\mathrm{GL}}_1$.