TRIPLE POINTS OF UNKNOTTING DISKS AND THE ARF INVARIANT OF KNOTS

An unknotting disc is the 'trace' in R4 of a homotopy of a diagram of a knot in R3, which shrinks the diagram to a point. In this paper we s tudy unknotting discs which have as singularities only ordinary triple points. It turns out that the Arf invariant of the knot is determined by the number of triple points in which all three branches of the dis c intersect pairwise with the same index. We call such a triple point coherent. This interpretation of the Arf invariant has a surprising co nsequence: Let S subset-of R4 be a taut immersed sphere which has as s ingularities only ordinary triple points. Then the number of coherent triple points in S is even. For example, it is easy to show that there is a taut immersed sphere S with Euler number six of the normal bundl e and which has exactly three ordinary double points and no other sing ularities. So, our result implies that the three double points of S ca n not be pushed together to create an ordinary triple point without th e appearance of new singularities. Here 'taut' means that the restrict ion of one of the coordinate functions on S has exactly two (non-degen erate) critical points, i.e. is a perfect Morse function.