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Holomorphic disks, link invariants and the multivariable Alexander polynomial, (2008)

by P Ozsváth, Z Szabó
Venue:Algebr. Geom. Topol.
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A combinatorial description of knot Floer homology

by Ciprian Manolescu, Peter Ozsváth, Sucharit Sarkar , 2006
"... Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of ..."
Abstract - Cited by 109 (30 self) - Add to MetaCart
Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.
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...21] and [25], this construction is extended to give an invariant, knot Floer homology ̂HFK, for null-homologous knots in a closed, oriented three-manifold. This construction is further generalized in =-=[23]-=- to the case of oriented links. The definition of all these invariants involves counts of holomorphic disks in the symmetric product of a Riemann surface, which makes them rather challenging to calcul...

Legendrian knots, transverse knots and combinatorial Floer homology

by Peter Ozsváth, Zoltán Szabó, Dylan Thurston , 2008
"... Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots (or links) in the three-sphere, with values in knot Floer homology. This invariant can also be used to construct an invariant of transverse knots. ..."
Abstract - Cited by 60 (8 self) - Add to MetaCart
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots (or links) in the three-sphere, with values in knot Floer homology. This invariant can also be used to construct an invariant of transverse knots.
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...O’s and X’s by {Oi} n i=1 and {Xi} n i=1 , and we denote the two sets by O and X, respectively. (We use here the notation from [MOST07]; the Oi correspond to the “white dots” of [MOS06] and the wi of =-=[OS05b]-=-, while the Xi to the “black dots” of [MOS06] and the zi of [OS05b].) Given a planar grid diagram G, we place it in a standard position on the plane by placing the bottom left corner at the origin, an...

Floer homology and surface decompositions

by András Juhász , 2006
"... Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if (M, γ) � (M ′ , γ ′ ) is a sutured manifold decomposition t ..."
Abstract - Cited by 35 (1 self) - Add to MetaCart
Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if (M, γ) � (M ′ , γ ′ ) is a sutured manifold decomposition then SFH(M ′ , γ ′ ) is a direct summand of SFH(M, γ). To prove the decomposition formula we give an algorithm that computes SFH(M, γ) from a balanced diagram defining (M, γ) that generalizes the algorithm of Sarkar and Wang. As a corollary we obtain that if (M, γ) is taut then SFH(M, γ) ̸ = 0. Other applications include simple proofs of a result of Ozsváth and Szabó that link Floer homology detects the Thurston norm, and a theorem of Ni that knot Floer homology detects fibred knots. Our proofs do not make use of any contact geometry or symplectic topology. Moreover, using these methods we show that if K is a genus g knot in a rational homology 3-sphere Y whose Alexander polynomial has leading coefficient ag ̸ = 0 and if rk̂HFK(Y, K, g) < 4 then Y \ N(K) admits a depth ≤ 1 taut foliation transversal to ∂N(K).
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... = 2, thus c(S, t0) = −2g(S). So we get that ⊕ SFH(Y (S)) = SFH(Y (K), s), s∈Spin c (Y (K)): 〈 c1(s,t0),[S] 〉=−2g(S) which in turn is isomorphic to ̂HFK(Y, K, [S], −g(S)) ≈ ̂HFK(Y, K, [S], g(S)), see =-=[7]-=-. Note that we get ̂HFK(Y, K, [S], g(S)) if we decompose along −S instead of S. □FLOER HOMOLOGY AND SURFACE DECOMPOSITIONS 23 Corollary 8.3. Let K be a knot in a rational homology 3-sphere Y whose Se...

Heegaard Floer homology and integer surgeries on links

by Ciprian Manolescu, Peter Ozsváth , 2011
"... ..."
Abstract - Cited by 20 (6 self) - Add to MetaCart
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THE DECATEGORIFICATION OF SUTURED FLOER HOMOLOGY

by Stefan Friedl, András Juhász, Jacob Rasmussen , 2009
"... We define a torsion invariant for balanced sutured manifolds and show that it agrees with the Euler characteristic of sutured Floer homology. The torsion is easily computed and shares many properties of the usual Alexander polynomial. ..."
Abstract - Cited by 17 (7 self) - Add to MetaCart
We define a torsion invariant for balanced sutured manifolds and show that it agrees with the Euler characteristic of sutured Floer homology. The torsion is easily computed and shares many properties of the usual Alexander polynomial.
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... component of ∂M has at least two sutures. We can reduce this case to the case treated above using Proposition [Ju08b, Proposition 9.2]. □ Remark. Link Floer homology of a link L ⊂ S 3 was defined in =-=[OS08a]-=-. It agrees with the sutured Floer homology of the sutured manifold S 3 (L) introduced in Example 2.3. In [OS08b] it is shown that if L has no trivial components then the link Floer homology of L dete...

Link Floer homology detects the Thurston norm

by Yi Ni , 2006
"... We prove that, for a link L in a rational homology 3–sphere, the link Floer homology detects the Thurston norm of its complement. This generalizes the previous results due to Ozsváth, Szabó and the author. ..."
Abstract - Cited by 16 (3 self) - Add to MetaCart
We prove that, for a link L in a rational homology 3–sphere, the link Floer homology detects the Thurston norm of its complement. This generalizes the previous results due to Ozsváth, Szabó and the author.
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...er result of the author. Our argument uses the techniques due to Ozsváth and Szabó, Hedden and the author. 57R58, 57M27; 57R30 1 Introduction Link Floer homology was introduced by Ozsváth and Szabó =-=[13]-=-, as a multifiltered theory for links in rational homology 3–spheres. This theory generalizes an earlier invariant for knots, the knot Floer homology (see Ozsváth and Szabó [11] and Rasmussen [15])....

GRID DIAGRAMS AND HEEGAARD FLOER INVARIANTS

by Ciprian Manolescu, Peter S. Ozsváth, Dylan P. Thurston , 2009
"... We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2Z). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also g ..."
Abstract - Cited by 16 (4 self) - Add to MetaCart
We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2Z). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsváth-Szabó mixed invariants of closed four-manifolds, in terms of grid diagrams.
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...tinguish homeomorphic four-manifolds with different smooth structures. In a different direction, there also exist Heegaard Floer invariants for null-homologous knots and links in three-manifolds (see =-=[17, 24, 22]-=-), which have applications to knot theory. One feature shared by the Donaldson, Seiberg-Witten, and Heegaard Floer invariants is that their original definitions are based on counting solutions to some...

LINK FLOER HOMOLOGY AND THE THURSTON NORM

by Peter Ozsváth, Zoltán Szabó , 2007
"... Abstract. We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To illustrate these techniques, we also compute the Thurs ..."
Abstract - Cited by 12 (0 self) - Add to MetaCart
Abstract. We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To illustrate these techniques, we also compute the Thurston polytopes of several specific link complements. 1.
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...e-sphere, this invariant is a bigraded Abelian group, whose graded Euler characteristic is the Alexander polynomial. Moreover, in this case, knot Floer homology detects the genus of the knot [21]. In =-=[25]-=-, the constructions from knot Floer homology are generalized to the case of links in S3 . For an ℓ-component link, this gives a multi-graded Abelian group, with one grading for each component of the l...

Heegaard Floer homology of broken fibrations over the circle. arXiv.org:0903.1773

by Yanki Lekili
"... This article is the first in a series where we investigate the relations between Perutz’s Lagrangian matching invariants and Ozsváth-Szabó’s Heegaard Floer invariants of three and four manifolds. In this paper, we deal with the purely Heegaard Floer theoretical side of this programme and prove an is ..."
Abstract - Cited by 10 (0 self) - Add to MetaCart
This article is the first in a series where we investigate the relations between Perutz’s Lagrangian matching invariants and Ozsváth-Szabó’s Heegaard Floer invariants of three and four manifolds. In this paper, we deal with the purely Heegaard Floer theoretical side of this programme and prove an isomorphism of 3–manifold invariants for certain spin c structures where the groups involved can be formulated in the language of Heegaard Floer theory. As applications, we give new calculations of Heegaard Floer homology of certain classes of 3–manifolds and a proof of Floer’s excision theorem in the context of Heegaard Floer homology. 57M50; 57R17 1
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...duct of moduli spaces M(Aleft) ×B M(Aright), where B = Sym nw (D) × Sym nw (D) and the fibre product is taken with respect to the above evaluation maps. This is a consequence of a gluing theorem (see =-=[18]-=- Theorem 5.1 for the proof in a very closely related situation and [3] for a discussion of gluing in a general context). Finally, we will prove that (eva right , evb right ) : M(Aa right ) × M(Ab righ...

Manifolds with small Heegaard Floer ranks

by Matthew Hedden, Yi Ni , 906
"... We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov ..."
Abstract - Cited by 8 (4 self) - Add to MetaCart
We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floer homology of the branched double cover, our results show that Khovanov homology detects the unknot if and only if it detects the two component unlink. 1
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... of papers, Ozsváth and Szabó defined invariants for a wide variety of topological and geometric objects in low dimensions, including threeand four- manifolds, knots and links, and contact structures =-=[23, 24, 26, 32, 27]-=-. These invariants proved to be quite powerful, with striking applications to questions in Dehn surgery, contact and symplectic geometry, knot concordance, and questions about unknotting numbers (to n...

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