Topologically and rationally slice knots

TL;DR

This paper demonstrates that the smooth concordance group of knots that are both topologically and rationally slice contains an infinite subgroup, revealing new complexity in the structure of such knots beyond previously known examples.

Contribution

It proves the existence of an infinite subgroup in the concordance group of topologically and rationally slice knots, showing richer algebraic structure than previously known.

Findings

The concordance group of topologically and rationally slice knots contains a  subgroup.

There are infinitely many topologically slice knots that are strongly rationally slice but not slice.

Previously known examples of such knots were only of order two.

Abstract

A knot in $S^3$ is topologically slice if it bounds a locally flat disk in $B^4$. A knot in $S^3$ is rationally slice if it bounds a smooth disk in a rational homology ball. We prove that the smooth concordance group of topologically and rationally slice knots admits a $\mathbb{Z}^\infty$ subgroup. All previously known examples of knots that are both topologically and rationally slice were of order two. As a direct consequence, it follows that there are infinitely many topologically slice knots that are strongly rationally slice but not slice.