  A geometric characterization of the Arf invariant of a knot in the 3-sphere
is given in terms of two kinds of 4-dimensional bordisms, half-gropes and
Whitney towers. These types of bordisms have associated complexities class and
order which filter the condition of bordism by an embedded annulus, i.e. knot
concordance, and it is shown constructively that the Arf invariant is exactly
the obstruction to cobording pairs of knots by half-gropes and Whitney towers
of arbitrarily high class and order. This illustrates geometrically how, in the
setting of knot concordance, the Vassiliev (isotopy) invariants "collapse" to
the Arf invariant.
