A generalization of the Tutte polynomials

TL;DR

This paper introduces a new generalization of Tutte polynomials for higher genus, proving they are complete matroid invariants, extending their known properties from genus one.

Contribution

It defines Tutte polynomials of genus g and proves they are complete matroid invariants, expanding the theoretical framework of matroid invariants.

Findings

Tutte polynomials of genus g are matroid invariants

They generalize known Tutte polynomials of genus one

Tutte polynomials of genus g are complete invariants

Abstract

In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants.