Sliced skein algebras and geometric Kauffman bracket

TL;DR

This paper studies sliced skein algebras of surfaces, calculating their centers and PI-degrees at roots of unity, and explores their properties and modules related to 3-manifold representations, revealing new algebraic and geometric insights.

Contribution

It establishes that sliced skein algebras are domains, computes their centers and PI-degrees at roots of unity, and introduces rho-reduced skein modules with applications to 3-manifold representations.

Findings

Sliced skein algebra of a finite type surface is a domain if the ground ring is a domain.

Centers and PI-degrees of sliced skein algebras are computed at roots of unity.

rho-reduced skein modules have dimension 1 for closed manifolds with irreducible representations.

Abstract

The sliced skein algebra of a closed surface of genus $g$ with $m$ punctures, $\mathfrak{S}=\Sigma_{g,m}$, is the quotient of the Kauffman bracket skein algebra $\mathcal{S}_\xi(\mathfrak{S})$ corresponding to fixing the scalar values of its peripheral curves. We show that the sliced skein algebra of a finite type surface is a domain if the ground ring is a domain. When the quantum parameter $\xi$ is a root of unity we calculate the center of the sliced skein algebra and its PI-degree. Among applications we show that any smooth point of a sliced character variety is a fully Azumaya point of the skein algebra $\mathcal{S}_\xi(\mathfrak{S})$. For any $SL_2(\mathbb{C})$--representation $\rho$ of the fundamental group of an oriented connected 3-manifold $M$ and a root of unity $\xi$ with odd $ord(\xi^2)$, we introduce the $\rho$-reduced skein module $\mathcal{S}_{\xi,\rho}(M)$. We show that $\mathcal{S}_{\xi,\rho}(M)$ has dimension 1 when $M$ is closed and $\rho$ is irreducible. We also show that if $\rho$ is irreducible the $\rho$-reduced skein module of a handlebody, as a module over the skein algebra of its boundary, is simple and has the dimension equal to the PI-degree of the skein algebra of its boundary.