A State Sum for the Total Face Color Polynomial

ABSTRACT The total face color polynomial is based upon the Poincaré polynomials of a family of filtered ‐color homologies. It counts the number of ‐face colorings of ribbon graphs for each positive integer . As such, it may be seen as a successor of the Penrose polynomial, which at 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.