Tutte Chromatic Identities and the Temperley-lieb Algebra

One of the remarkable features 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 show that this result fits in the framework of quantum topology and give a proof of Tutte’s identity using the notion of the chromatic algebra, whose Markov trace is the chromatic polynomial of an associated graph. We also show that another relation of Tutte’s for the chromatic polynomial at Q = φ + 1 precisely corresponds to a Jones-Wenzl projector in the Temperley-Lieb algebra. We show that such a relation exists whenever Q = 2 + 2 cos(2πj/(n + 1)) for j < n positive integers. When j = 1, these are the Beraha numbers, and in this case the existence of such a relation was conjectured by Tutte. We present a recursive formula for this sequence of chromatic polynomial relations.