Skein modules and character varieties of Seifert manifolds

TL;DR

This paper investigates the structure of skein modules and character varieties of Seifert fibered 3-manifolds, establishing conditions for finite generation and computing skein modules, linking their dimensions to character variety sizes.

Contribution

It provides a characterization of when the skein module of a Seifert manifold is finitely generated and computes these modules, connecting them to character variety properties.

Findings

Skein modules are finitely generated iff the manifold is irreducible and non-Haken.

Character varieties are shown to be reduced under mild conditions.

Skein module dimensions match the size of the character variety.

Abstract

We show that the Kauffman bracket skein module of a closed Seifert fibered 3-manifold $M$ is finitely generated over $\mathbb Z[A^{\pm 1}]$ if and only if $M$ is irreducible and non-Haken. We analyze in detail the character varieties $X(M)$ of such manifolds and show that under mild conditions they are reduced. We compute the Kauffman bracket skein modules for these $3$-manifolds (over $\mathbb Q(A)$) and show that their dimensions coincide with $|X(M)|.$