Tutte Chromatic Identities from the Temperley-lieb Algebra

One remarkable feature of the chromatic polynomial χ(Q) is Tutte’s golden identity. This relates χ(φ+ 2) for any triangulation of the sphere to (χ(φ+ 1)) for the same graph, where φ denotes the golden ratio. We explain how this result fits in the framework of quantum topology and give a proof using the chromatic algebra, whose Markov trace is the chromatic polynomial of an associated graph. We also explain how this identity is a consequence of level-rank duality for SO(N) topological quantum field theories and Birman-Murakami-Wenzl algebras. We then show that another identity of Tutte’s for the chromatic polynomial at Q = φ + 1 arises from a Jones-Wenzl projector in the Temperley-Lieb algebra. We generalize this identity to each value Q = 2 + 2 cos(2πj/(n + 1)) for j < n positive integers. When j = 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.