Casson towers and slice links

TL;DR

This paper proves new results about the existence of embedded discs in Casson towers of certain heights and constructs new slice knots and links, providing insights into the slice-ribbon conjecture.

Contribution

It establishes that Casson towers of height 4 contain embedded discs and develops methods to find slicing discs for certain knots and links.

Findings

Casson tower of height 4 contains a flat embedded disc.

Disc embedding results for height 2 and 3 Casson towers.

Construction of new slice knots and links, including potential counterexamples to the homotopy ribbon slice conjecture.

Abstract

We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we 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 proofs we create a capped grope from a Casson tower and use a refined height raising argument to establish the existence of a symmetric grope which has two layers of caps, data which is sufficient for a topological disc to exist, with the desired boundary. 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.