Virtual knot cobordism and the affine index polynomial

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner.

On spanning surfaces of links

In his paper on knot cobordism groups in codimension 2, Levine develops conditions for a knotted Sn in Sn+2 to bound a disc in Bn+3 In this paper some of his methods are extended to introduce a necessary condition for a classical link in S3 to bound a surface of specified genus in B4. In particular, this answers a question of Zeemann's about some links related to the ‘Mazur link’.

Cobordism of theta curves in S3

In this paper we show that the cobordism classes of theta curves in S3 form a group under vertex connected sum. We investigate this group by means of knot cobordism and link cobordism.

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

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.