Skein module dimensions of mapping tori of the 2-torus

We determine the dimension of the Kauffman bracket skein module at generic $q$ for mapping tori of the 2-torus, generalising 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 $G={\mathrm{S}\mathrm{L}}_N$ or ${\mathrm{G}\mathrm{L}}_N$, which is a direct summand of the whole skein module, and from which the dimensions follow easily in the cases $G={\mathrm{S}\mathrm{L}}_2$ and $G={\mathrm{G}\mathrm{L}}_1$.