Kauffman bracket versus Jones polynomial skein modules

TL;DR

This paper compares two types of skein modules, those based on Kirby-Melvin relations and Kauffman bracket relations, providing insights and examples relevant to quantum topology and Chern-Simons theory.

Contribution

It establishes a detailed comparison between Kirby-Melvin skein modules and Kauffman bracket skein modules, clarifying their relationship and differences.

Findings

Identifies conditions under which the skein modules are equivalent.

Provides explicit examples illustrating the differences.

Discusses implications for quantum invariants and topological quantum field theories.

Abstract

This paper resolves the problem of comparing the skein modules defined using the skein relations discovered by R. Kirby and P. Melvin that underlie the Reshetikhin-Turaev model for $SU(2)$ Chern-Simons theory to the Kauffman bracket skein modules. Several applications and examples are presented.