Rank-expanding satellite operators on the topological knot concordance group

TL;DR

This paper shows that certain satellite operations on knots can expand the rank in the topological concordance group, extending previous smooth category results to the topological setting.

Contribution

It demonstrates the existence of rank-expanding satellite operators in the topological concordance group, a phenomenon previously known only in the smooth category.

Findings

Existence of rank-expanding satellite operators in the topological concordance group.

Such examples do not exist in the algebraic concordance group.

Extension of smooth category results to the topological setting.

Abstract

Given a fixed knot P in a solid torus and any knot K in S^3, one can form the satellite of K with pattern P. This operation induces a self-map of the concordance group of knots in S^3. It has been proved by Dai, Hedden, Mallick, and Stoffregen that in the smooth category there exist P for which this function is rank-expanding; that is, for some K, the set {P(nK)} generates an infinite rank subgroup. Here we demonstrate that similar examples exist in the case of the topological locally flat concordance group. Such examples cannot exist in the algebraic concordance group.