The Penrose-Kauffman Polynomial

Louis H. Kauffman; Daniel S. Silver; Susan G. Williams

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arXiv · 2604.16635

The Penrose-Kauffman Polynomial

Louis H. Kauffman·Daniel S. Silver·Susan G. Williams

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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.

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Louis H. Kauffman, Daniel S. Silver, Susan G. Williams. 2026-04-17. The Penrose-Kauffman Polynomial. https://arxiv.org/abs/2604.16635

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