The skein partition function of the mapping torus

TL;DR

This paper calculates the dimensions of GL_N-skein modules for genus-one mapping tori, introducing a skein partition function with an explicit Euler product expansion, advancing understanding of quantum topology invariants.

Contribution

It provides a new explicit computation of the skein partition function for genus-one mapping tori with arbitrary diffeomorphisms and generic quantum parameters.

Findings

Derived explicit formulas for skein module dimensions.

Introduced the skein partition function with Euler product expansion.

Enhanced understanding of quantum invariants of 3-manifolds.

Abstract

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.