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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 185 (32 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 135 (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 threemanifold invariants: properties and applications
"... ̂HF(Y, s),and HFred(Y, s) associated to closed, oriented threemanifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with SeibergWitten theory. The pr ..."
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Cited by 128 (27 self)
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̂HF(Y, s),and HFred(Y, s) associated to closed, oriented threemanifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with SeibergWitten theory. 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. 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 118 (18 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.
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 95 (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.
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 68 (8 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 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 62 (10 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.
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 62 (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.
Holomorphic triangle invariants and the topology of symplectic fourmanifolds
 Duke Math. J
"... This article analyzes the interplay between symplectic geometry in dimension 4 and the invariants for smooth fourmanifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic fourmanifolds, which leads to new ..."
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Cited by 40 (4 self)
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This article analyzes the interplay between symplectic geometry in dimension 4 and the invariants for smooth fourmanifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic fourmanifolds, which leads to new proofs of the indecomposability theorem for symplectic fourmanifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of fourmanifolds along a certain class of threemanifolds obtained by plumbings of spheres. This leads to restrictions on the topology of Stein fillings of such threemanifolds.
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 36 (0 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.