Evaluations of topological Tutte polynomials

TL;DR

This paper explores properties of the topological transition polynomial $Q(G)$ of embedded graphs, revealing new relations and interpretations among various graph polynomials like $P(G)$, $R(G)$, and $T(G)$, and reformulating classical results such as the Four Colour Theorem.

Contribution

It introduces new properties of $Q(G)$, explains similarities among polynomials, and provides combinatorial interpretations and a reduction formula for the tensor product of embedded graphs.

Findings

Expressed $P(G)$, $R(G)$, and $T(G)$ as sums of chromatic polynomials.

Showed $R(G)$ counts edge 3-colourings.

Reformulated the Four Colour Theorem in terms of $R(G)$.

Abstract

We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollob\'as and Riordan's ribbon graph polynomial, $R(G)$, and the topological Penrose polynomial, $P(G)$. The general framework provided by $Q(G)$ also leads to several other combinatorial interpretations these polynomials. In particular, we express $P(G)$, $R(G)$, and the Tutte polynomial, $T(G)$, as sums of chromatic polynomials of graphs derived from $G$; show that these polynomials count $k$-valuations of medial graphs; show that $R(G)$ counts edge 3-colourings; and reformulate the Four Colour Theorem in terms of $R(G)$. We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of $P(G)$ and $R(G)$.