A Penrose polynomial for embedded graphs

TL;DR

This paper generalizes the Penrose polynomial to graphs embedded in arbitrary surfaces, revealing new identities, relations, and combinatorial interpretations, and offers a novel reformulation of the Four Colour Theorem.

Contribution

It extends the Penrose polynomial to embedded graphs, introduces new identities, and connects it with other polynomials and dualities, providing fresh insights and a reformulation of the Four Colour Theorem.

Findings

Derived a deletion-contraction relation for the Penrose polynomial.

Connected the Penrose polynomial to the circuit partition polynomial.

Reformulated the Four Colour Theorem using chromatic polynomials of twisted duals.

Abstract

We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which can not be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable checkerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the Four Colour Theorem.