Invariants of Tutte Partitions and a $q$-Analogue

TL;DR

This paper introduces a new construction of the Tutte polynomial for matroids and q-matroids using Tutte partitions, establishing a q-analogue and invariance properties, and connecting the rank and Tutte polynomials.

Contribution

It develops a novel approach to defining the Tutte polynomial via Tutte partitions and introduces a q-analogue with invariance properties, extending classical concepts.

Findings

Tutte polynomial constructed from Tutte partitions for matroids and q-matroids.

Established a q-analogue of Tutte-Grothendiek invariance.

Connected the rank polynomial and Tutte polynomial through convolution.

Abstract

We describe a construction of the Tutte polynomial for both matroids and $q$-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition. We show that such partitions in the matroid case include the class of partitions arising in Crapo's definition of the Tutte polynomial, while not representing a direct $q$-analogue of such partitions. We propose axioms of $q$-Tutte-Grothendiek invariance and show that this yields a $q$-analogue of Tutte-Grothendiek invariance. We establish the connection between the rank polynomial and the Tutte polynomial, showing that one can be obtained from the other by convolution.