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

• Published in: Journal of topology 2024-12, Vol.17 (4), p.e70008-n/a
• Main Authors: Dai, Irving, Hedden, Matthew, Mallick, Abhishek, Stoffregen, Matthew
• Format: Article
• Language: English
• ISSN: 1753-8416; 1753-8424; 1753-8424
• DOI: 10.1112/topo.70008

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 Pinzón‐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 that 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 τ$\tau$‐invariant. Our results also recover and extend several theorems in this area established using instanton Floer homology.