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arXiv · 2211.11666

Invariants of Tutte Partitions and a $q$-Analogue

Eimear Byrne·Andrew Fulcher
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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.

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Eimear Byrne, Andrew Fulcher. 2022-11-21. Invariants of Tutte Partitions and a $q$-Analogue. https://arxiv.org/abs/2211.11666

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