Kauffman bracket skein module of the $(3,3,3,3)$-pretzel link exterior

TL;DR

This paper demonstrates that the Kauffman bracket skein module of a specific pretzel link exterior is not finitely generated, providing a counterexample to a previously conjectured finiteness property in knot theory.

Contribution

It proves that the skein module for the (3,3,3,3)-pretzel link exterior over a certain field is not finitely generated, disproving Detcherry's 2021 conjecture.

Findings

Skein module is not finitely generated over the specified ring.

Disproves the finiteness conjecture of Detcherry (2021).

Provides new insights into the structure of skein modules for complex links.

Abstract

We show that the Kauffman bracket skein module of the $(3,3,3,3)$-pretzel link exterior over $\mathbb{Q}(q^{\frac{1}{2}})$ is not finitely generated as a module over $\mathbb{Q}(q^{\frac{1}{2}})[t_1,t_2]$, where $t_1,t_2$ are the meridians of two components. This disproves a finiteness conjecture of Detcherry proposed in 2021.