Rank-expanding satellites, Whitehead doubles, and Heegaard Floer homology

TL;DR

This paper demonstrates that many satellite operators are rank-expanding in the concordance group, providing new Floer-theoretic criteria and extending previous results with Whitehead doubles and instanton Floer homology.

Contribution

It systematically studies the rank-expanding property of satellite operators, introduces Floer-theoretic conditions, and extends known theorems to a broader class of patterns and companions.

Findings

Many satellite operators are rank-expanding.

Established Floer-theoretic conditions for infinite-rank images.

Constructed an infinite family of Whitehead doubles with negative τ-invariant.

Abstract

We show that a large class of satellite operators are rank-expanding; that is, they map some rank-one subgroup of the concordance group onto an infinite linearly independent set. Our work constitutes the first systematic study of this property in the literature and partially affirms a conjecture of the second author and Pinz\'on-Caicedo. More generally, we establish a Floer-theoretic condition for a family of companion knots to have infinite-rank image under satellites from this class. The methods we use are amenable to patterns which act trivially in topological concordance and are capable of handling a surprisingly wide variety of companions. For instance, we give an infinite linearly independent family of Whitehead doubles whose companion knots all have negative $\tau$-invariant. Our also results recover and extend several theorems in this area established using instanton Floer homology.