Skein Algebras of Three-Manifolds at 4th Roots of Unity

Charles Frohman; Joanna Kania-Bartoszynska; Thang Le

doi:10.48550/arxiv.2212.04558

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Skein Algebras of Three-Manifolds at 4th Roots of Unity

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Skein Algebras of Three-Manifolds at 4th Roots of Unity

Charles Frohman, Joanna Kania-Bartoszynska and Thang Le

ArXiv.org

12/08/2022

DOI: 10.48550/arxiv.2212.04558

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Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

This paper introduces an algebra structure on the part of the skein module of an arbitrary $3$-manifold $M$ spanned by links that represent $0$ in $H_1(M;\mathbb{Z}_2)$ when the value of the parameter used in the Kauffman bracket skein relation is equal to $\pm {\bf i}$. It is proved that if $M$ has no $2$-torsion in $H_1(M;\mathbb{Z})$ then those algebras, $K_{\pm {\bf i}}^0(M)$, are naturally isomorphic to the corresponding algebras when the value of the parameter is $\pm 1$. This implies that the algebra $K_{\pm{\bf i}}^0(M)$ is the unreduced coordinate ring of the variety of $PSL_2(\mathbb{C})$-characters of $\pi_1(M)$ that lift to $SL_2(\mathbb{C})$-representations.

Details

Title: Subtitle: Skein Algebras of Three-Manifolds at 4th Roots of Unity
Creators: Charles Frohman
Joanna Kania-Bartoszynska
Thang Le
Resource Type: Preprint
Publication Details: ArXiv.org
DOI: 10.48550/arxiv.2212.04558
ISSN: 2331-8422
Language: English
Date posted: 12/08/2022
Academic Unit: Mathematics
Record Identifier: 9984327057602771

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