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42
A combinatorial description of knot Floer homology
, 2006
"... Given a grid presentation of a knot (or link) K in the threesphere, 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 ..."
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Cited by 109 (30 self)
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Given a grid presentation of a knot (or link) K in the threesphere, 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.
Legendrian knots, transverse knots and combinatorial Floer homology
, 2008
"... Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots (or links) in the threesphere, with values in knot Floer homology. This invariant can also be used to construct an invariant of transverse knots. ..."
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Cited by 60 (8 self)
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Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots (or links) in the threesphere, with values in knot Floer homology. This invariant can also be used to construct an invariant of transverse knots.
Floer homology and surface decompositions
, 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 ..."
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Cited by 35 (1 self)
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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 3sphere 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).
THE DECATEGORIFICATION OF SUTURED FLOER HOMOLOGY
, 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. ..."
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Cited by 17 (7 self)
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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.
Link Floer homology detects the Thurston norm
, 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. ..."
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Cited by 16 (3 self)
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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.
GRID DIAGRAMS AND HEEGAARD FLOER INVARIANTS
, 2009
"... We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary threemanifolds (with coefficients in Z/2Z). The descriptions are based on presenting the threemanifold as an integer surgery on a link in the threesphere, and then using a grid diagram for the link. We also g ..."
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Cited by 16 (4 self)
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We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary threemanifolds (with coefficients in Z/2Z). The descriptions are based on presenting the threemanifold as an integer surgery on a link in the threesphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 OzsváthSzabó mixed invariants of closed fourmanifolds, in terms of grid diagrams.
LINK FLOER HOMOLOGY AND THE THURSTON NORM
, 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 multivariable Alexander polynomial. To illustrate these techniques, we also compute the Thurs ..."
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Cited by 12 (0 self)
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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 multivariable Alexander polynomial. To illustrate these techniques, we also compute the Thurston polytopes of several specific link complements. 1.
Heegaard Floer homology of broken fibrations over the circle. arXiv.org:0903.1773
"... This article is the first in a series where we investigate the relations between Perutz’s Lagrangian matching invariants and OzsváthSzabó’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 ..."
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Cited by 10 (0 self)
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This article is the first in a series where we investigate the relations between Perutz’s Lagrangian matching invariants and OzsváthSzabó’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
Manifolds with small Heegaard Floer ranks
, 906
"... We show that the only irreducible threemanifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zeroframed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov ..."
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Cited by 8 (4 self)
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We show that the only irreducible threemanifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zeroframed 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