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Abstract We prove a formula for the involutive concordance invariants of the cabled knots in terms of those of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer homology.
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Rank‐expanding satellites, Whitehead doubles, and Heegaard Floer homology
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 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 ‐invariant. Our results also recover and extend several theorems in this area established using instanton Floer homology.
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- PAR ID:
- 10556913
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of Topology
- Volume:
- 17
- Issue:
- 4
- ISSN:
- 1753-8416
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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