Link invariants, the chromatic polynomial and the Potts model

Abstract

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 χ 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.

Tutte chromatic identities from the Temperley–Lieb algebra

“…Readers familiar with the Potts model and/or the Tutte polynomial may recall that it has long been known (indeed from Temperley and Lieb's original paper [21]) that the chromatic polynomial and the more general Tutte polynomial can be related to the Markov trace of elements of the Temperley-Lieb algebra. The relation derived in [7] and here is quite different from the earlier relation; in statistical-mechanics language ours arises from the low-temperature expansion of the Potts model, while the earlier one arises from Fortuin-Kasteleyn cluster expansion [6].…”

“…Our companion paper [6] discusses these connections in detail. In particular, the relation between the SO.3/ Birman-Wenzl-Murakami algebra and the chromatic algebra described initially by the first author and Read [7] is derived there.…”

“…An important and rather natural tool for analyzing the chromatic polynomial relations is the chromatic algebra C Q n introduced (in a different context) by Martin and Woodcock [16], and analyzed in depth by the authors in [6]. The basic idea in the definition of the chromatic algebra is to consider the contraction-deletion rule as a linear relation in the vector space spanned by graphs, rather than just a relation defining the chromatic polynomial.…”

“…Tutte conjectured in [24] that there is such a relation, similar to (1)(2) at C 1 D B 5 , for each Beraha number, and we give a recursive formula for these relations based on the formula for the Jones-Wenzl projectors P .n/ . In fact, based on the relation between the chromatic algebra and the SO.3/ BMW algebra [6], it seems reasonable to conjecture that these are all linear relations which preserve the chromatic polynomial at a given value of Q (and in particular there are no such relations at Q not equal to one of special values (1-4)).…”

Anyonic quantum spin chains: Spin-1 generalizations and topological stability

“…(33) The projectors P (0) and P (1) project two neighboring spin-1 loops onto the spin-0 and spin-1 channels, respectively (see 50 for explicit expressions of these projectors in terms of the BMW algebra). The number of Potts states Q is related to the quantum dimension of the spin-1/2 anyons d 1/2 (or the parameter d appearing in the Temperley-Lieb algebra) via Q = d 2 1/2 = 4 cos(π/(k + 2)) 2 , and thus Q = 1, 2, 3, 4 corresponds to k = 1, 2, 4, ∞.…”