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112
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 150 (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 117 (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.
Absolutely graded Floer homologies and intersection forms for fourmanifolds with boundary
 Advances in Mathematics 173
, 2003
"... Abstract. In [22], we introduced absolute gradings on the threemanifold invariants developed in [21] and [20]. Coupled with the surgery long exact sequences, we obtain a number of three and fourdimensional applications of this absolute grading including strengthenings of the “complexity bounds ” ..."
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Cited by 101 (26 self)
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Abstract. In [22], we introduced absolute gradings on the threemanifold invariants developed in [21] and [20]. Coupled with the surgery long exact sequences, we obtain a number of three and fourdimensional applications of this absolute grading including strengthenings of the “complexity bounds ” derived in [20], restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF + for a variety of threemanifolds. Moreover, we show how the structure of HF + constrains the exoticness of definite intersection forms for smooth fourmanifolds which bound a given threemanifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson’s diagonalizability theorem and the Thom conjecture for CP 2. 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 100 (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.
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 83 (13 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.
Holomorphic triangles and invariants for smooth fourmanifolds
"... Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed fourmanifolds, built using the Floer homology theories defined in [8] and [12]. This fourdimensional theory also endows the corresponding threedimensional theories with additional structure: an absolute gradi ..."
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Cited by 76 (24 self)
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Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed fourmanifolds, built using the Floer homology theories defined in [8] and [12]. This fourdimensional theory also endows the corresponding threedimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone of these constructions is the study of holomorphic disks in the symmetric products of Riemann surfaces. 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 62 (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.
Heegaard Floer homology and alternating knots
, 2002
"... In [23] we introduced a knot invariant for a nullhomologous knot K in an oriented threemanifold Y, which is closely related to the Heegaard Floer homology of Y (c.f. [21]). In this paper we investigate some properties of these knot homology groups for knots in the threesphere. We give a combinato ..."
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Cited by 60 (16 self)
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In [23] we introduced a knot invariant for a nullhomologous knot K in an oriented threemanifold Y, which is closely related to the Heegaard Floer homology of Y (c.f. [21]). In this paper we investigate some properties of these knot homology groups for knots in the threesphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (compare [24]). Applications include new restrictions on the Alexander polynomial of alternating knots.
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 58 (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 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 55 (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 nonzero 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 fourball genera of knots admitting lens space surgeries (which are sharp for Berge’s knots), and a constraint on threemanifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts. 1.