The Penrose-Kauffman Polynomial

TL;DR

This paper introduces the Penrose-Kauffman polynomial for cubic graphs on surfaces, linking it to chromatic polynomials and knot theory, and relates the Four Color Theorem to link diagram colorings.

Contribution

It provides a new polynomial invariant for cubic graphs on surfaces and connects classical graph coloring problems to knot theory and link diagram colorings.

Findings

The Penrose-Kauffman polynomial sums chromatic polynomials of associated graphs.

Knot-theoretic methods yield elementary proofs of polynomial properties.

The Four Color Theorem is equivalent to a coloring statement about link diagrams.

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.