A state sum for the total face color polynomial

Scott Baldridge; Louis H. Kauffman; Ben McCarty

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arXiv · 2308.02732

A state sum for the total face color polynomial

Scott Baldridge·Louis H. Kauffman·Ben McCarty

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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.

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Scott Baldridge, Louis H. Kauffman, Ben McCarty. 2023-08-04. A state sum for the total face color polynomial. https://arxiv.org/abs/2308.02732

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