Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The skein partition function of the mapping torus

Julia Bierent, David Jordan, Matthias Vancraeynest, Monica Vazirani

TL;DR

This work computes the dimensions of GL_N skein modules for genus-one mapping tori by formulating a skein partition function that encodes all ranks N. The authors recast the problem in terms of twisted Hochschild homology of the GL_N skein category and reduce the computation to the quantum torus, ultimately translating it into an orbit-counting problem for centralizers in the symmetric group. They obtain an explicit Euler product for the skein partition function, $\mathcal{Z}_{M_\gamma}(t)=\prod_{k\ge1}(1-t^k)^{-c_k(\gamma)}$, with $c_k(\gamma)$ given by a number-theoretic style formula in terms of $\operatorname{tr}(\gamma)$ and concrete special cases (shears, negatives, and identity). The paper discusses positivity of the exponents $c_k(\gamma)$ and situates the results within a broader Donaldson-Thomas/invariant framework, conjecturing deeper connections to derived skein modules and BPS cohomology. Overall, the results provide a concrete, computable enumerative framework for GL_N skein dimensions of mapping tori and open avenues toward a DT-skein correspondence and generalizations to other classical groups.

Abstract

We compute the dimensions of $\text{GL}_N$-skein modules of genus-one mapping tori $T^2\times_γS^1$, 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.

The skein partition function of the mapping torus

TL;DR

This work computes the dimensions of GL_N skein modules for genus-one mapping tori by formulating a skein partition function that encodes all ranks N. The authors recast the problem in terms of twisted Hochschild homology of the GL_N skein category and reduce the computation to the quantum torus, ultimately translating it into an orbit-counting problem for centralizers in the symmetric group. They obtain an explicit Euler product for the skein partition function, , with given by a number-theoretic style formula in terms of and concrete special cases (shears, negatives, and identity). The paper discusses positivity of the exponents and situates the results within a broader Donaldson-Thomas/invariant framework, conjecturing deeper connections to derived skein modules and BPS cohomology. Overall, the results provide a concrete, computable enumerative framework for GL_N skein dimensions of mapping tori and open avenues toward a DT-skein correspondence and generalizations to other classical groups.

Abstract

We compute the dimensions of -skein modules of genus-one mapping tori , for an arbitrary diffeomorphism of , and for generic quantum parameter. These are most cleanly expressed via a generating function over all , which we dub the skein partition function, and for which we compute an explicit Euler product expansion.

Paper Structure

This paper contains 14 sections, 9 theorems, 58 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The skein partition function
  3. Positivity and relation to Donaldson-Thomas invariants
  4. Outline of the paper
  5. Acknowledgements
  6. Preliminaries
  7. GLN skein theory
  8. Symmetric group conventions
  9. Mapping torus and sl2Z
  10. Skein module of the mapping torus via Hochschild homology
  11. Hochschild homology of the skein category via the quantum torus
  12. Dimension of the skein module
  13. Twisted Hochschild homology expressed as a cokernel
  14. Computing coinvariants of the twisted Hochschild homology via orbit counting

Key Result

Theorem 1.2

Let $\gamma \in \operatorname{SL}_{2}(\mathbb{Z})$. The skein partition function $\mathop{\mathrm{\mathcal{Z}}}\nolimits_{M_\gamma}(t)$ of the mapping torus $M_\gamma$ admits an Euler product expansion, where the $c_k(\gamma)$ are all positive integers determined by the trace $\mathop{\mathrm{tr}}\nolimits(\gamma)$. More precisely, we have: in the case that $\mathop{\mathrm{tr}}\nolimits(\gamma)

Figures (1)

  • Figure 1: Overview of the results for all possible conjugacy classes of $\gamma\in\operatorname{SL}_{2}(\mathbb{Z})$.

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Remark 1.8
  • Remark 1.9
  • Conjecture 1.10
  • ...and 27 more