The skein partition function of the mapping torus

We compute the dimensions of ${\text{GL}}_N$-skein modules of genus-one mapping tori $T^2{\times }_{\gamma }{S}^1$, for an arbitrary diffeomorphism of $T^2$, and for generic quantum parameter. These are most cleanly expressed via a generating function over all $N$, which we dub the skein partition function, and for which we compute an explicit Euler product expansion.