Constructing a Tutte polynomial for graphs embedded in surfaces

TL;DR

This paper explores different methods to extend the Tutte polynomial to graphs embedded in surfaces, demonstrating their equivalence and providing an accessible introduction to the topic.

Contribution

It introduces three distinct approaches to defining the Tutte polynomial for embedded graphs and proves their equivalence, linking to known polynomials like the Bollobás-Riordan polynomial.

Findings

All three approaches yield the same polynomial.

The polynomial generalizes the Tutte polynomial to surface-embedded graphs.

Provides a foundational understanding of Tutte polynomials in topological graph theory.

Abstract

There are several different extensions of the Tutte polynomial to graphs embedded in surfaces. To help frame the different options, here we consider the problem of extending the Tutte polynomial to cellularly embedded graphs starting from first principles. We offer three different routes to defining such a polynomial and show that they all lead to the same polynomial. This resulting polynomial is known in the literature under a few different names including the ribbon graph polynomial, and 2-variable Bollobas-Riordan polynomial. Our overall aim here is to use this discussion as a mechanism for providing a gentle introduction to the topic of Tutte polynomials for graphs embedded in surfaces.