Skein module dimensions of mapping tori of the 2-torus

Patrick Kinnear

doi:10.4171/QT/231

Skein module dimensions of mapping tori of the 2-torus

Geometric Topology 2025-12-19 v4 Quantum Algebra

Authors: Patrick Kinnear

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Abstract

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 $G = \mathrm{SL}_N$ 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 $G = \mathrm{GL}_1$ .

Keywords

skein module mapping torus 2-torus kauffman bracket hochschild homology sl_n gl_n

Cite

@article{arxiv.2304.07332,
  title  = {Skein module dimensions of mapping tori of the 2-torus},
  author = {Patrick Kinnear},
  journal= {arXiv preprint arXiv:2304.07332},
  year   = {2025}
}

Comments

v4 adds a remark commenting on a simplification of the main formula that did not appear in the published version; v3 text as published in Quantum Topology; v2 corrects an error in the main theorem; v1 original manuscript

Skein module dimensions of mapping tori of the 2-torus

Abstract

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