Non-surjective satellite operators and piecewise-linear concordance

TL;DR

This paper constructs specific satellite knots that are not smoothly slice in any homology 4-ball, revealing new insights into knot concordance and the structure of homology spheres.

Contribution

It introduces a particular pattern in satellite operations that produces knots with non-slice properties in all homology 4-balls, advancing understanding of knot concordance.

Findings

Existence of a knot pattern producing non-slice satellite knots

Construction of a knot in a homology sphere without a PL disk in any homology 4-ball

Implications for the structure of homology spheres and knot concordance

Abstract

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-linear disk in any homology 4-ball.