A Tutte polynomial for maps II: the non-orientable case

TL;DR

This paper introduces a new polynomial invariant for maps on surfaces, unifying several existing polynomials and extending their applicability to non-orientable surfaces, with implications for counting flows and tensions.

Contribution

It constructs a comprehensive polynomial invariant for maps on both orientable and non-orientable surfaces, generalizing multiple known polynomials and their extensions.

Findings

Unifies various polynomial invariants into a single framework

Extends the Tutte polynomial to non-orientable surfaces

Enables counting of local flows and tensions in finite groups

Abstract

We construct a new polynomial invariant of maps (graphs embedded in a compact surface, orientable or non-orientable), which contains as specializations the Krushkal polynomial, the Bollob\'as--Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial. Other evaluations include the number of local flows and local tensions taking non-identity values in a given finite group.