Dimer face polynomials in knot theory and cluster algebras

TL;DR

This paper explores the connections between dimer face polynomials in bipartite graphs, knot theory, and cluster algebras, revealing new combinatorial interpretations and algebraic relationships.

Contribution

It establishes that dimer face polynomials generalize Alexander polynomials and are F-polynomials in cluster algebras, linking combinatorics, knot theory, and algebraic structures.

Findings

Dimer face polynomials relate to Alexander polynomials of links.

Dimer face polynomials are F-polynomials in cluster algebras.

Nonvanishing Plücker coordinates are cluster monomials.

Abstract

The set of perfect matchings of a connected bipartite plane graph $G$ has the structure of a distributive lattice, as shown by Propp, where the partial order is induced by the height of a matching. In this article, our focus is the dimer face polynomial of $G$, which is the height generating function of all perfect matchings of $G$. We connect the dimer face polynomial on the one hand to knot theory, and on the other to cluster algebras. We show that certain dimer face polynomials are multivariate generalizations of Alexander polynomials of links, highlighting another combinatorial view of the Alexander polynomial. We also show that an arbitrary dimer face polynomial is an $F$-polynomial in the cluster algebra whose initial quiver is dual to the graph $G$. As a result, we recover a recent representation theoretic result of Bazier-Matte and Schiffler that connects $F$-polynomials and Alexander polynomials, albeit from a very different, dimer-based perspective. As another application of our results, we also show that all nonvanishing Pl\"ucker coordinates on open positroid varieties are cluster monomials.