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Holomorphic Disks and Topological Invariants for Closed ThreeManifolds
 ANN. OF MATH
, 2000
"... The aim of this article is to introduce certain topological invariants for closed, oriented threemanifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y  U0 tie U1, these theories are variants of the Lagrangian Floer homology for the gfold symmetric product of Y relat ..."
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Cited by 278 (35 self)
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The aim of this article is to introduce certain topological invariants for closed, oriented threemanifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y  U0 tie U1, these theories are variants of the Lagrangian Floer homology for the gfold symmetric product of Y relative to certain totally real subspaces associated to U0 and U1.
Floer homology and knot complements
, 2003
"... Abstract. We use the OzsváthSzabó theory of Floer homology to define an invariant of knot complements in threemanifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the OzsváthSzabó Floer homology of large integral surgeries on the k ..."
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Cited by 237 (7 self)
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Abstract. We use the OzsváthSzabó theory of Floer homology to define an invariant of knot complements in threemanifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the OzsváthSzabó Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of perfect knots in S3 for which ĈF r has a particularly simple form. For these knots, formal properties of the OzsváthSzabó theory enable us to make a complete calculation of the Floer homology. It turns out that most small knots are perfect. 1.
Holomorphic disks and knot invariants
 Adv. in Math
, 2004
"... Abstract. We define a Floerhomology invariant for knots in an oriented threemanifold, closely related to the Heegaard Floer homologies for threemanifolds defined in [18]. We set up basic properties of these invariants, including an Euler characteristic calculation, behaviour under connected sums. ..."
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Cited by 182 (20 self)
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Abstract. We define a Floerhomology invariant for knots in an oriented threemanifold, closely related to the Heegaard Floer homologies for threemanifolds defined in [18]. We set up basic properties of these invariants, including an Euler characteristic calculation, behaviour under connected sums. Then, we establish a relationship with HF + for surgeries along the knot. Applications include calculation of HF + of threemanifolds obtained by surgeries on some special knots in S 3, and also calculation of HF + for certain simple threemanifolds which fiber over the circle. 1.
HOLOMORPHIC DISKS AND THREEMANIFOLD INVARIANTS: PROPERTIES AND APPLICATIONS
, 2001
"... ... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. The properties include a ..."
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Cited by 177 (28 self)
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... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. The properties include a relationship between the Euler characteristics of HF ± and Turaev’s torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to threemanifold topology.
Heegaard Floer homologies and contact structures
 Duke Math. J
"... Abstract. Given a contact structure on a closed, oriented threemanifold Y, we describe an invariant which takes values in the threemanifold’s Floer homology ̂ HF (in the sense of [10]). This invariant vanishes for overtwisted contact structures and is nonzero for Stein fillable ones. The construc ..."
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Cited by 132 (12 self)
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Abstract. Given a contact structure on a closed, oriented threemanifold Y, we describe an invariant which takes values in the threemanifold’s Floer homology ̂ HF (in the sense of [10]). This invariant vanishes for overtwisted contact structures and is nonzero for Stein fillable ones. The construction uses of Giroux’s interpretation of contact structures in terms of open book decompositions (see [4]), and the knot Floer homologies introduced in [14]. 1.
On the Heegaard Floer homology of branched doublecovers
 Adv. Math
"... Abstract. Let L ⊂ S 3 be a link. We study the Heegaard Floer homology of the branched doublecover Σ(L) of S 3, branched along L. When L is an alternating link, ̂HF of its branched doublecover has a particularly simple form, determined entirely by the determinant of the link. For the general case, ..."
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Cited by 114 (11 self)
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Abstract. Let L ⊂ S 3 be a link. We study the Heegaard Floer homology of the branched doublecover Σ(L) of S 3, branched along L. When L is an alternating link, ̂HF of its branched doublecover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E 2 term is a suitable variant of Khovanov’s homology for the link L, converging to the Heegaard Floer homology of Σ(L). 1.
Knot Floer Homology and the fourball genus
 Geom. Topol
"... Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotti ..."
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Cited by 96 (9 self)
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Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the fourball genera of torus knots. As another illustration, we use calculate the invariant for several tencrossing knots. 1.
On the Floer homology of plumbed threemanifolds
 Geom. Topol
"... Abstract. We calculate HF + for threemanifolds obtained by plumbings of spheres specified by certain graphs. Our class of graphs is sufficiently large to describe, for example, all Seifert fibered rational homology spheres. These calculations can be used to determine also the Floer homology of othe ..."
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Cited by 90 (8 self)
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Abstract. We calculate HF + for threemanifolds obtained by plumbings of spheres specified by certain graphs. Our class of graphs is sufficiently large to describe, for example, all Seifert fibered rational homology spheres. These calculations can be used to determine also the Floer homology of other threemanifolds, including the product of a circle with a genus two surface. 1.
Knot Floer homology detects genusone 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 genusone knots and links, obtaining as a corollary that if ra ..."
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Cited by 75 (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 genusone 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 lefthanded trefoil knot.
On knot Floer homology and cabling
, 2005
"... Abstract. In this paper we study the knot Floer homology groups ̂HFK(S 3, K2,n), where K2,n denotes the (2, n) cable of an arbitrary knot, K. It is shown that for sufficiently large n, the Floer homology of the cabled knot depends only on the filtered chain homotopy type of ĈFK(K). In fact, the ho ..."
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Cited by 55 (9 self)
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Abstract. In this paper we study the knot Floer homology groups ̂HFK(S 3, K2,n), where K2,n denotes the (2, n) cable of an arbitrary knot, K. It is shown that for sufficiently large n, the Floer homology of the cabled knot depends only on the filtered chain homotopy type of ĈFK(K). In fact, the homology groups in the top 2 filtration dimensions for the cabled knot are isomorphic to the original knot’s Floer homology group in the top filtration dimension. The results are extended to (p, pn±1) cables. As an example we compute ̂HFK((T2,2m+1)2,2n+1) for all sufficiently large n, where T2,2m+1 denotes the (2, 2m + 1)torus knot. 1.