A state sum for the total face color polynomial

TL;DR

This paper introduces a state sum formula for the total face color polynomial, connecting topological quantum field theory and diagrammatic tensor approaches to graph coloring.

Contribution

It presents a novel state sum formula for the total face color polynomial, unifying topological quantum field theory and diagrammatic tensor perspectives.

Findings

Provides a new computational method for the total face color polynomial.

Unifies two different theoretical approaches to graph coloring.

Extends understanding of face colorings in ribbon graphs.

Abstract

The total face color polynomial is based upon the Poincar\'{e} polynomials of a family of filtered $n$-color homologies. It counts the number of $n$-face colorings of ribbon graphs for each positive integer $n$. As such, it may be seen as a successor of the Penrose polynomial, which at $n=3$ counts $3$-edge colorings (and consequently $4$-face colorings) of planar trivalent graphs. In this paper we describe a state sum formula for the polynomial. This formula unites two different perspectives about graph coloring: one based upon topological quantum field theory and the other on diagrammatic tensors.