A state sum for the total face color polynomial

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.