On the evaluation at (-i,i) of the Tutte polynomial of a binary matroid

TL;DR

This paper explores the complex evaluation of the Tutte polynomial at (-i, i) for binary matroids, revealing its dependence on a quadratic form and providing polynomial-time evaluation methods.

Contribution

It introduces a new understanding of the argument of T_M(-i, i) via a canonical quadratic form and offers algorithms for polynomial-time computation of related invariants.

Findings

Expression of the argument in terms of a quadratic form

Polynomial-time evaluation algorithm for T_M(-i, i)

Description of the canonical tripartition and related invariants

Abstract

Vertigan has shown that if $M$ is a binary matroid, then $|T_M(-\iota,\iota)|$, the modulus of the Tutte polynomial of $M$ as evaluated in $(-\iota, \iota)$, can be expressed in terms of the bicycle dimension of $M$. In this paper, we describe how the argument of the complex number $T_M(-\iota,\iota)$ depends on a certain $\zfour$-valued quadratic form that is canonically associated with $M$. We show how to evaluate $T_M(-\iota,\iota)$ in polynomial time, as well as the canonical tripartition of $M$ and further related invariants.