A basis for the Kauffman skein module of the product of a surface and a circle

Detcherry, Renaud; Wolff, Maxime

doi:10.2140/agt.2021.21.2959

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A basis for the Kauffman skein module of the product of a surface and a circle

Renaud Detcherry and Maxime Wolff

Algebraic & Geometric Topology 21 (2021) 2959–2993

DOI: 10.2140/agt.2021.21.2959

Abstract

The Kauffman bracket skein module $𝒮 (M)$ of a $3$ –manifold $M$ is a $ℚ (A)$ –vector space spanned by links in $M$ modulo the so-called Kauffman relations. For any closed oriented surface $Σ$ we provide an explicit spanning family for the skein modules $𝒮 (Σ \times S^{1})$ . Combined with earlier work of Gilmer and Masbaum (Proc. Amer. Math. Soc. 147 (2019) 4091–4106), we answer their question about the dimension of $𝒮 (Σ \times S^{1})$ being $2^{2 g + 1} + 2 g - 1$ .

Keywords

knot theory, skein modules, quantum topology

Mathematical Subject Classification 2010

Primary: 57M27

References

Publication

Received: 27 January 2020

Revised: 10 July 2020

Accepted: 5 August 2020

Published: 22 November 2021

Authors

Renaud Detcherry
Max Planck Institute for Mathematics Bonn Germany	Institut de Mathématiques de Bourgogne Université de Bourgogne Dijon France
http://detcherry.perso.math.cnrs.fr/
Maxime Wolff
Université Pierre et Marie Curie - Paris 6 Institut de Mathématiques de Jussieu Paris France
https://webusers.imj-prg.fr/~maxime.wolff/

Volume 21, issue 6 (2021)