Link invariants, the chromatic polynomial and the Potts model

Abstract

We study the connections between link invariants, the chromatic polynomial, geometric representations of models of statistical mechanics, and their common underlying algebraic structure. We establish a relation between several algebras and their associated combinatorial and topological quantities. In particular, we define the chromatic algebra, whose Markov trace is the chromatic polynomial $χQ$ of an associated graph, and we give applications of this new algebraic approach to the combinatorial properties of the chromatic polynomial. In statistical mechanics, this algebra occurs in the low-temperature expansion of the $Q$-state Potts model. We establish a relationship between the chromatic algebra and the $SO(3)$ Birman–Murakami–Wenzl algebra, which is an algebra-level analogue of the correspondence between the $SO(3)$ Kauffman polynomial and the chromatic polynomial.