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

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.