Computing the Tutte Polynomial of a Matroid from its Lattice of Cyclic Flats

TL;DR

This paper presents a method to compute the Tutte polynomial of a matroid using its lattice of cyclic flats, revealing that the polynomial is determined by this lattice and related statistics, thus advancing matroid theory understanding.

Contribution

It introduces a way to derive the Tutte polynomial from the lattice of cyclic flats, generalizing previous results and aiding in classifying matroids with identical Tutte polynomials.

Findings

Tutte polynomial can be derived from cyclic flats lattice

Matroid Tutte polynomial is determined by lattice structure and ranks

Generalizes results for perfect matroid designs

Abstract

We show how the Tutte polynomial of a matroid $M$ can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats. The results imply that the Tutte polynomial of $M$ is already determined by the abstract lattice of its cyclic flats together with their cardinalities and ranks. They furthermore generalize a similiar statement for perfect matroid designs due to Mphako and help to understand families of matroids with identical Tutte polynomial as constructed by Ken Shoda.