Link cobordism and the intersection of slice discs

TL;DR

This paper investigates the slice properties of 2-links in 4-spheres, proving that for any 2-component 2-link bounding a 5-ball, there exist embedded discs with transverse intersections forming trivial knots.

Contribution

It establishes the existence of embedded discs for 2-component 2-links with controlled intersections, advancing understanding of link slicing in 4-dimensional topology.

Findings

Existence of embedded discs for 2-component 2-links in 4-spheres

Discs intersect transversely with trivial knot intersections

Progress towards understanding slice properties of 2-links

Abstract

It is well-known that all 2-knots are slice. Are all 2-links slice? This is an outstanding open question. In this paper we prove the following: For any 2-component 2-link (J,K)in the 4-sphere which bounds the 5-ball B^5, there is an embedded disc 2-disc D^2_J (respectively, D^2_K) in B^5 with the following properties: J (respectively K) bounds D^2_J (respectively, D^2_K). D^2_J and D^2_K intersect transversely. the intersection of D^2_J and D^2_K in D^2_J (respectively, D^2_K) is a trivial 1-knot.