On knot Floer homology and lens space surgeries

2003/03/02 by Peter Ozsváth, Peter Ozsvath, Zoltán Szabó +3 · 11 citations
Mathematics · Medicine · #53D #57M #57R #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Symplectic Geometry (math.SG) #math.GT #math.SG #msc:53D #msc:57M #msc:57R

paper · pdf · doi:10.48550/arxiv.math/0303017

24 pages, 2 figures

openalex publication_date 2003/03/02 · arxiv created 2004/03/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In an earlier paper, we used the absolute grading on Heegaard Floer homology to give restrictions on knots in S³ which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that all the non-zero coefficients of the Alexander polynomial of such a knot are ± 1. This information in turn can be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L(p,q) which arise as integral surgeries on knots in S³ with |p|≤ 1500. Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge's knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts.

Cited by

• Homology cobordism for some Sol manifolds
• On the Alexander polynomials of modular knots
• SL₂(\mathbb R)-representations and left-orderable surgeries of (-2, 3, 2n+1)-pretzel knots
• Deforming abelian elliptic SL(2,ℝ)--representations of knot groups
• On 3-manifolds admitting co-orientable taut foliations, but none with vanishing Euler class
• Colored knot Floer homology: structures and examples
• A note on real Heegaard Floer homology and localization
• The bridge index and the braid index for twist positive knots
• Ribbon knots and iterated cables of fibered knots
• Non-braid positive hyperbolic L-space knots
• Bordered contact invariants and half Giroux torsion

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