Link invariants, the chromatic polynomial and the Potts model

Details Link invariants, the chromatic polynomial and the Potts model Link invariants, the chromatic polynomial and the Potts model

2008-06-20

arxiv.org/abs/0806.3484v3

Adv. Theor. Math. Phys. 14 (2010), 507-540

Mathematics

Combinatorics

25 pages. v2: section on level-rank duality moved to arXiv:0711.0016 v3: references updated

Scientific paper

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.

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