Simple Whitney Towers, Half-gropes and the Arf Invariant of a Knot

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, respectively. This illustrates geometrically how, in the setting of knot concordance, the Vassiliev (isotopy) invariants ‘collapse’ to the Arf invariant.