The action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the 3-twist knot complement

TL;DR

This paper explicitly describes how the Kauffman bracket skein algebra of a torus acts on the skein module of a 3-twist knot complement, linking knot theory, quantum algebra, and Chern-Simons theory.

Contribution

It provides a detailed example of the action of the skein algebra on a knot complement's skein module, connecting knot invariants with quantum field theory concepts.

Findings

Explicit computation of the skein algebra action

Connection to noncommutative A-polynomial

Insight into quantum systems in Chern-Simons theory

Abstract

We determine the action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the complement of the 3-twist knot. The point is to study the relationship between knot complements and their boundary tori, an idea that has proved very fruitful in knot theory. We place this idea in the context of Chern-Simons theory, where such actions arose in connection with the computation of the noncommutative version of the A-polynomial that was defined by Frohman, Gelca and Lofaro, but they can also be interpreted as quantum mechanical systems. Our goal is to exhibit a detailed example in a part of Chern-Simons theory where examples are scarce.