Link invariants, the chromatic polynomial and the Potts model

TL;DR

This paper explores the deep connections between link invariants, the chromatic polynomial, and the Potts model, introducing a new algebraic framework that unifies these concepts and reveals their underlying algebraic structures.

Contribution

It defines the chromatic algebra and links it to statistical mechanics models and link invariants, establishing new algebraic relationships and applications.

Findings

Introduces the chromatic algebra with a Markov trace equal to the chromatic polynomial.

Establishes a relationship between the chromatic algebra and the SO(3) BMW algebra.

Provides applications to combinatorial properties of the chromatic polynomial.

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 \chi_Q 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.