Some comments concerning the Levine approach to slicing classical knots (Q1092459)

In higher dimensions \((n>1)\), a (2n-1)-dimensional knot K in \(S^{2n+1}\) is slice only if every submanifold of \(S^{2n+1}\) bounded by K can be ``surgered'' to a disk in \(D^{2n+2}\) (\({\mathcal L}\)-compressible). This paper examines the situation for \(n=1\). Although it still seems that there is no example of a Seifert surface for a slice knot which is known not to be \({\mathcal L}\)-compressible, several positive results are obtained here. For example it is pointed out that, for any fibred knot K, any Seifert surface of K{\#}(-K) is \({\mathcal L}\)-compressible. The main result is: If K is a ribbon knot, then any Seifert surface of K which does not intersect the self-intersection arcs of some ribbon disk of K is \({\mathcal L}\)-compressible. One interesting consequence is that any Seifert surface of a ribbon knot can be made \({\mathcal L}\)-compressible by doing ambient 1-surgeries.