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

TL;DR

This paper studies skein algebras of 3-manifolds at 4th roots of unity, revealing their structure and relation to character varieties, especially under certain homological conditions.

Contribution

It introduces a new algebra structure on skein modules at 4th roots of unity and establishes isomorphisms with character variety coordinate rings under specific conditions.

Findings

Algebra structure on skein modules at 4th roots of unity.

Isomorphism with character variety coordinate rings when no 2-torsion.

Identification of these algebras with unreduced PSL_2(C)-character varieties.

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.