Kauffman bracket polynomials, perfect matchings and cluster variables

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.