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46
On knot Floer homology and lens space surgery
"... Abstract. In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 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 th ..."
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Cited by 140 (13 self)
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Abstract. In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 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 S 3 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. 1.
Knot Floer homology detects genus-one fibred links
, 2008
"... Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai’s theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots and links, obtaining as a corollary that if ra ..."
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Cited by 79 (1 self)
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Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai’s theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots and links, obtaining as a corollary that if rational surgery on a knot K gives the Poincaré homology sphere Σ(2, 3, 5), then K is the left-handed trefoil knot.
Contact structures with distinct Heegaard-Floer invariants
"... Abstract. We prove that the contact structures on Y = ∂X induced by nonhomotopic Stein structures on the 4-manifold X have distinct Heegaard Floer invariants. 1. ..."
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Cited by 37 (3 self)
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Abstract. We prove that the contact structures on Y = ∂X induced by nonhomotopic Stein structures on the 4-manifold X have distinct Heegaard Floer invariants. 1.
OZSVÁTH-SZABÓ INVARIANTS AND FILLABILITY OF CONTACT STRUCTURES
, 2004
"... Recently, P. Ozsváth and Z. Szabó defined an invariant of contact structures with values in the Heegaard-Floer homology groups. They also proved that the twisted invariant of a weakly symplectically fillable contact structures is non trivial. In this article we prove with an example that their non v ..."
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Cited by 30 (6 self)
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Recently, P. Ozsváth and Z. Szabó defined an invariant of contact structures with values in the Heegaard-Floer homology groups. They also proved that the twisted invariant of a weakly symplectically fillable contact structures is non trivial. In this article we prove with an example that their non vanishing result does not hold in general for the untwisted contact invariant. As a consequence of this fact, we show how Heegaard-Floer theory can distinguish between weakly and strongly symplectically fillable contact structures.
Holomorphic disks and genus bounds
- Geom. Topol
"... Abstract. We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the genus of a knot. This leads to some new proofs of certain results previously obtained usin ..."
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Cited by 28 (8 self)
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Abstract. We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the genus of a knot. This leads to some new proofs of certain results previously obtained using Seiberg-Witten monopole Floer homology (in collaboration with Kronheimer and Mrowka). It also leads to a purely Morse-theoretic interpretation of the genus of a knot. The method of proof shows that the canonical element of Heegaard Floer homology associated to a weakly symplectically fillable contact structure is non-trivial. In particular, for certain three-manifolds, Heegaard Floer homology gives obstructions to the existence of taut foliations. 1.
Heegaard Floer homology and fibred 3–manifolds
- Amer. J. of Math
"... Given a closed 3–manifold Y, we show that the Heegaard Floer homology determines whether Y fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier result about knots proved by Ghiggini and the author. 1 ..."
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Cited by 22 (6 self)
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Given a closed 3–manifold Y, we show that the Heegaard Floer homology determines whether Y fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier result about knots proved by Ghiggini and the author. 1
On the topology of broken Lefschetz fibrations and near-symplectic four-manifolds
, 2007
"... The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subc ..."
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Cited by 18 (3 self)
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The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefschetz fibrations/pencils, while showing that all near-symplectic closed 4-manifolds can be supported by these à la Auroux, Donaldson, Katzarkov. Various constructions of broken Lefschetz fibrations and a generalization of the symplectic fiber sum operation to the near-symplectic setting are given. Extending the study of Lefschetz fibrations, we detect certain constraints on the symplectic fiber sum operation to result in a 4-manifold with nontrivial Seiberg-Witten invariant, as well as the self-intersection numbers that sections of broken Lefschetz fibrations can acquire.
LATTICE COHOMOLOGY OF NORMAL SURFACE SINGULARITIES
, 2007
"... For any negative definite plumbed 3–manifold M we construct from its plumbed graph a graded Z[U]–module. This, for rational homology spheres, conjecturally equals the Heegaard–Floer homology of Ozsváth and Szabó, but it has even more structure. If M is a complex singularity link then the normalized ..."
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Cited by 17 (6 self)
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For any negative definite plumbed 3–manifold M we construct from its plumbed graph a graded Z[U]–module. This, for rational homology spheres, conjecturally equals the Heegaard–Floer homology of Ozsváth and Szabó, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic can be compared with the analytic invariants. The Seiberg–Witten Invariant Conjecture of [16, 13] is discussed in the light of this new object.
