Not all knots are smoothly round handle slice

TL;DR

This paper demonstrates that while many knots are topologically round handle slice, infinitely many knots do not possess this property smoothly, highlighting a distinction between topological and smooth categories in 4-manifold topology.

Contribution

The paper provides the first examples of knots that are not smoothly round handle slice, establishing a clear difference between topological and smooth sliceness.

Findings

Infinitely many knots are not smoothly round handle slice.

Topological and smooth sliceness properties differ significantly.

Supports the conjecture that smooth structures are more restrictive than topological ones.

Abstract

Freedman and Krushkal showed that if the surgery conjecture and the $s$-cobordism conjecture hold for all topological 4-manifolds, then every link with pairwise zero linking numbers is topologically round handle slice. Kim, Powell, and Teichner showed that every knot is topologically round handle slice. We show that infinitely many knots fail to be smoothly round handle slice.