A 2-torsion invariant of 2-knots

In this paper we describe what should perhaps be called a `type-2' Vassiliev invariant of knots S^2 -> S^4. We give a formula for an invariant of 2-knots, taking values in Z_2 that can be computed in terms of the double-point diagram of the knot. The double-point diagram is a collection of curves and diffeomorphisms of curves, in the domain S^2, that describe the crossing data with respect to a projection, analogous to a chord diagram for a projection of a classical knot S^1 -> S^3. Our formula turns the computation of the invariant into a planar geometry problem. More generally, we describe a numerical invariant of families of knots S^j -> S^n, for all n >= j+2 and j >= 1. In the co-dimension two case n=j+2 the invariant is an isotopy invariant, and either takes values in Z or Z_2 depending on a parity issue.