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 $\chi \left(Q\right)$ 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 $SO\left(N\right)$ topological quantum field theories and Birman–Murakami–Wenzl algebras. This identity is a remarkable feature of the chromatic polynomial relating $\chi \left(\varphi +2\right)$ for any triangulation of the sphere to ${\left(\chi \left(\varphi +1\right)\right)}^2$ for the same graph, where $\varphi$ 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 $Q=\varphi +1$ arises from a Jones–Wenzl projector in the Temperley–Lieb algebra. We generalize this identity to each value $Q=2+2cos\left(2\pi j∕\left(n+1\right)\right)$ 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.