Casson towers and slice links

Jae Choon Cha, Mark Powell

Published 2014-11-06Version 1

We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle. We also prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group assumptions. In the proof we create a capped grope from a Casson tower and use a refined height raising argument to establish the existence of a symmetric grope of height 4 which has two layers of caps, data which is sufficient for a topological disc with the desired boundary to exist. As applications, we present new slice knots and links by giving direct geometric constructions of slicing discs. In particular we construct a family of slice knots which are potential counterexamples to the homotopy ribbon slice conjecture.