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Triple points of unknotting discs and the Arf invariant of knots

Published online by Cambridge University Press:  24 October 2008

Thomas Fiedler
Thomas Fiedler
Affiliation:
Laboratoire de Topologie et Géométrie, URA CNRS 1408, Université Paul Sabatier, 118, route de Narbonne, 31061 Toulouse Cédex, France
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Abstract

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 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 point coherent. This interpretation of the Arf invariant has a surprising consequence:

Let S ⊂ ℝ4 be a taut immersed sphere which has as singularities 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 bundle and which has exactly three ordinary double points and no other singularities. So, our result implies that the three double points of S can 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 on S has exactly two (non-degenerate) critical points, i.e. is a perfect Morse function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 1 , July 1994 , pp. 119 - 129
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

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