Ribbon fibred knots, cobordism of surface diffeomorphisms, and pseudo-Anosov diffeomorphisms

Abstract

For a long time, the main activity in knot theory, we would even say the only one for the problems related to knot cobordism, has been focused onto the development and the analysis of various algebraic invariants. The present paper intends to illustrate some geometric techniques, and to advertise a recent theorem of Casson and Gordon (8) which provides a necessary condition for a fibred classical knot to be ribbon (see definition below) in terms of a cobordism property of its monodromy. We want to show how this last result, combined with some previous work of ours on the cobordism of surface diffeomorphisms (4) (see also (10)) and Thurston's theory of pseudo-Anosov diffeomorphisms (28), (12), can effectively be used to show that a knot is not ribbon.