The Penrose-Kauffman Polynomial
Geometric Topology
2026-04-21 v1
Abstract
For any cubic graph in a closed orientable surface and a perfect matching, the Penrose-Kauffman polynomial is a sum of chromatic polynomials of a collection of associated graphs. A knot-theoretic perspective affords elementary proofs of old and new results about the polynomial. The Four Color Theorem is shown to be equivalent to a statement about 3-coloring alternating link diagrams in the plane that are reduced and have no bigon regions.
Cite
@article{arxiv.2604.16635,
title = {The Penrose-Kauffman Polynomial},
author = {Louis H. Kauffman and Daniel S. Silver and Susan G. Williams},
journal= {arXiv preprint arXiv:2604.16635},
year = {2026}
}
Comments
33 pages, 28 figures