Tutte chromatic identities from the Temperley–Lieb algebra

Abstract

This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial .Q/ of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chromatic polynomial. In particular, we show that Tutte's golden identity is a consequence of level-rank duality for SO.N / topological quantum field theories and BirmanMurakami-Wenzl algebras. This identity is a remarkable feature of the chromatic polynomial relating . C 2/ for any triangulation of the sphere to . . C 1// 2 for the same graph, where denotes the golden ratio. The new viewpoint presented here explains that Tutte's identity is special to these values of the parameter Q. A natural context for analyzing such properties of the chromatic polynomial is provided by the chromatic algebra, whose Markov trace is the chromatic polynomial of an associated graph. We use it to show that another identity of Tutte's for the chromatic polynomial at Q D C 1 arises from a Jones-Wenzl projector in the Temperley-Lieb algebra. We generalize this identity to each value Q D 2 C 2 cos.2 j =.n C 1// for j < n positive integers. When j D 1, these Q are the Beraha numbers, where the existence of such identities was conjectured by Tutte. We present a recursive formula for this sequence of chromatic polynomial relations.57M15; 05C15, 57R56, 81R05

Link invariants, the chromatic polynomial and the Potts model

“…We give an algebraic proof of Tutte's golden identity for the chromatic polynomial in a companion publication [10]. This striking non-linear identity plays a very interesting role in describing quantum loop models of "Fibonacci anyons", where it implies that these loop models should yield topological quantum field theories in the continuum limit [24,8,12,14].…”

“…Equivalently, one may start with the space of trivalent graphs, modulo the relations in figure 9, see theorem 4.3 in the next section. The chromatic algebra C Q (more precisely, its generalization, the chromatic category, see section 4 in [10]) is a local version of V Q 2 , i.e. it corresponds to Σ =disk.…”

“…The trace pairing descends to a positive-definite Hermitian product on the quotient of C Bn by the trace radical (see corollary 7.2), which gives rise to a positive-definite Hermitian product on V Σ n , see [36]. (The trace radical is analyzed in [10], where we show that it contains the pull-back of the Jones Wenzl-projector from the Temperley-Lieb algebra, and give applications of this fact to the structure of the chromatic polynomial of planar graphs. )…”

“…As part of the proof, in this section we find an additive basis of the chromatic algebra C n in terms of planar partitions, and we establish an algebra analogue of the "state sum" formula (3.2) for the chromatic polynomial. Both the chromatic algebra and its trivalent presentation established here are used in our companion paper [10] …”

“…Quantum loop models provide lattice spin systems whose low-energy excitations in the continuum limit are described by a topological quantum field theory [14]. In both classical and quantum cases, algebraic relations such as the level-rank duality described in [10] allow one to map seemingly different loop models onto each other. This turns out to be quite useful in locating critical points in loop models [9], a matter of great importance for finding quantum loop models which describe topological order.…”