Polynomial invariants of graphs on surfaces

TL;DR

This paper explores the relationship between combinatorial parameters of embedded graphs and topological features of the surface, expressing a polynomial invariant as a specialization of the Krushkal polynomial using matroid theory.

Contribution

It establishes a connection between graph invariants, matroid parameters, and topological surface properties, extending the understanding of polynomial invariants in topological graph theory.

Findings

Expresses the polynomial as a specialization of the Krushkal polynomial

Relates combinatorial parameters of cycle and bond matroids to surface topology

Provides a new perspective on graph invariants in topological embeddings

Abstract

For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This will give an expression of the polynomial, defined by M.Las Vergnas in a combinatorial way using matroids as a specialization of the Krushkal polynomial, defined using the symplectic structure in the first homology group of the surface.