On the evaluation at (-ι,ι) 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 -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.