A Tutte polynomial for maps

TL;DR

This paper introduces the surface Tutte polynomial, a new invariant for maps that unifies several existing polynomials and encodes various combinatorial properties of maps embedded in orientable surfaces.

Contribution

It constructs the surface Tutte polynomial, generalizing previous polynomials and capturing diverse combinatorial invariants of maps in a unified framework.

Findings

Contains the Las Vergnas, Bollobás-Riordan, and Kruskhal polynomials as special cases.

Includes evaluations for the number of local tensions and flows in finite groups.

Encompasses the count of quasi-forests within its evaluations.

Abstract

We follow the example of Tutte in his construction of the dichromate of a graph (that is, the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, Bollob\'as-Riordan polynomial and Kruskhal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.