On Tutte's chromatic invariant

Consider a simple connected graph $\mathsf{G}$ embedded in the plane together with a contractible circuit $\mathsf{J}$. For a partition $\phi$ of the vertex set of $\mathsf{J}$ we denote by $P_{(\mathsf{G},\phi)}(t)$ the number of ways of assigning one of $t$ given colours to each vertex of $\mathsf{G}$ so that vertices in the same block of $\phi$ have the same colour. Tutte showed that this polynomial may be expressed uniquely as a linear combination of $P_{(\mathsf{G},\pi)}(t)$ over all planar partitions $\pi$ of $\mathsf{J}$, with scalars $\vartheta_{\phi,\pi}(t)$ that are independent of $\mathsf{G}$. We show that the (chromatic) invariants $\vartheta_{\phi,\pi}$ have a natural algebraic setting in terms of the orthogonal projection from the partition algebra $\mathbb{P}_r(t)$ to the Temperley-Lieb subalgebra $\mathbb{TL}_r(t,1)$. We define the genus of a partition and give an extension of the invariants to arbitrary genus $g$. Finally, we summarise the rôle of the genus 0 invariants in the algebraic approach of Birkhoff and Lewis to the Four Colour Theorem.