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 (??,?)|, the modulus of the Tutte polynomial of M as evaluated in (??,?), 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 (??,?) depends on a certain $\mathbb{Z}/4\mathbb {Z}$ -valued quadratic form that is canonically associated with M. We show how to evaluate T M (??,?) in polynomial time, as well as the canonical tripartition of M and further related invariants.