A note on rationally slice knots

TL;DR

This paper proves that all known rationally slice but not slice knots can be embedded in a single 4-manifold, independent of the specific knot, simplifying the understanding of their structure.

Contribution

It demonstrates that the rational homology ball bounding these knots is unique up to diffeomorphism, unifying previously separate examples into a single 4-manifold framework.

Findings

The rational homology ball V_K is independent of the knot K up to diffeomorphism.

All known rationally slice but not slice knots are contained in connected sums of a single 4-manifold.

The result simplifies the classification of such knots by unifying their bounding 4-manifolds.

Abstract

Kawauchi proved that every strongly negative amphichiral knot $K \subset S^3$ bounds a smoothly embedded disk in some rational homology ball $V_K$, whose construction a priori depends on $K$. We show that $V_K$ is independent of $K$ up to diffeomorphism. Thus, a single 4-manifold, along with connected sums thereof, accounts for all known examples of knots that are rationally slice but not slice.