Advances in Theoretical and Mathematical Physics
Volume 14 (2010)
Link invariants, the chromatic polynomial and the Potts model
Pages: 507 – 540
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.