Kauffman bracket polynomials, perfect matchings and cluster variables

TL;DR

This paper connects Kauffman bracket polynomials of certain links to perfect matchings in bipartite graphs and relates them to cluster algebra variables, expanding understanding of link invariants and their algebraic structures.

Contribution

It introduces a new class of links with bracket polynomials linked to perfect matchings and establishes their relation to cluster variables, generalizing previous results.

Findings

Bracket polynomials can be expanded over perfect matchings.

Links include 2-bridge, pretzel, and Montesinos links.

Bracket polynomials are specializations of F-polynomials in cluster theory.

Abstract

We introduce a class of links whose bracket polynomials admit an expansion over perfect matchings of a plane bipartite graph. This class includes 2-bridge links, pretzel links, and Montesinos links. Our first main result (Theorem A) provides a partial answer to a question posed by Kauffman concerning the connection between spanning tree expansions of the Jones polynomial and the Clock Theorem. Building on Theorem A, we apply our framework to cluster theory and prove in Theorem B that the bracket polynomials of links in this class can be realized as specializations of the F-polynomials of certain cluster variables. Theorem B generalizes several earlier results. We also present several applications and illustrative examples.