Triple points of unknotting discs and the Arf invariant of knots

Abstract

AbstractAn unknotting disc is the ‘trace’ in ℝ4of a homotopy of a diagram of a knot in ℝ3, which shrinks the diagram to a point. In this paper we study 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 disc intersect pairwise with the same index. We call such a triple pointcoherent. This interpretation of the Arf invariant has a surprising consequence:LetS⊂ ℝ4be a taut immersed sphere which has as singularities only ordinary triple points. Then the number of coherent triple points inSis even. For example, it is easy to show that there is a taut immersed sphereSwith Euler number six of the normal bundle and which has exactly three ordinary double points and no other singularities. So, our result implies that the three double points ofScan not be pushed together to create an ordinary triple point without the appearance of new singularities.Here ‘taut’ means that the restriction of one of the coordinate functions onShas exactly two (non-degenerate) critical points, i.e. is a perfect Morse function.