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Abstract We consider manifold-knot pairs$$(Y,K)$$, whereYis a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface$$\Sigma $$in a homology ballX, such that$$\partial (X, \Sigma ) = (Y, K)$$can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from$$(Y, K)$$to any knot in$$S^3$$can be arbitrarily large. The proof relies on Heegaard Floer homology.
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This content will become publicly available on January 1, 2026
Floer homology and non-fibered knot detection
Abstract We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots and that HOMFLY homology detects infinitely many knots; these theories were previously known to detect a mere six knots, all fibered. These results rely on our main technical theorem, which gives a complete classification of genus-1 knots in the 3-sphere whose knot Floer homology in the top Alexander grading is 2-dimensional. We discuss applications of this classification to problems in Dehn surgery which are carried out in two sequels. These include a proof that$$0$$-surgery characterizes infinitely many knots, generalizing results of Gabai from his 1987 resolution of the Property R Conjecture.
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- Award ID(s):
- 1952707
- PAR ID:
- 10598953
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 13
- ISSN:
- 2050-5086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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