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.