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28
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.
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 79 (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.
Knots with unknotting number one and Heegaard Floer homology
"... Abstract. We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal ..."
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Cited by 23 (3 self)
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Abstract. We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number equal to one. We also classify alternating knots with ten crossings and unknotting number equal to one. 1.
L–space surgeries, genus bound, and the cabling conjecture
"... Abstract. We establish a tight inequality relating the knot genus g(K) and the surgery slope p under the assumption that pframed Dehn surgery along K is an Lspace that bounds a sharp 4manifold. This inequality applies in particular when the surgered manifold is a lens space or a connected sum the ..."
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Cited by 19 (1 self)
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Abstract. We establish a tight inequality relating the knot genus g(K) and the surgery slope p under the assumption that pframed Dehn surgery along K is an Lspace that bounds a sharp 4manifold. This inequality applies in particular when the surgered manifold is a lens space or a connected sum thereof. Combined with work of GordonLuecke, Hoffman, and MatignonSayari, it follows that if surgery along a knot produces a connected sum of lens spaces, then the knot is either a torus knot or a cable thereof, confirming the cabling conjecture in this case. 1.
Khovanov homology, open books, and tight contact structures
"... Abstract. We define the reduced Khovanov homology of an open book (S, φ), and we identify a distinguished “contact element ” in this group which may be used to establish the tightness of contact structures compatible with (S, φ). Our construction generalizes the relationship between the reduced Khov ..."
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Cited by 18 (3 self)
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Abstract. We define the reduced Khovanov homology of an open book (S, φ), and we identify a distinguished “contact element ” in this group which may be used to establish the tightness of contact structures compatible with (S, φ). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Steinfillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined in [5]. 1.
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 and genus one, one boundary component open books
, 2008
"... We compute the Heegaard Floer homology of any rational homology 3sphere with an open book decomposition of the form (T, φ), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of every T²bundle over S¹ with first Betti number equal to on ..."
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Cited by 11 (3 self)
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We compute the Heegaard Floer homology of any rational homology 3sphere with an open book decomposition of the form (T, φ), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of every T²bundle over S¹ with first Betti number equal to one, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Steinfillings of the contact structures compatible such open books, to narrow down somewhat the class of 3braid knots with finite concordance order, and to identify all quasialternating links with braid index at most 3.
Lens space surgeries and a conjecture of Goda and Teragaito
, 2004
"... Abstract. Using work of Ozsváth and Szabó, we show that if a nontrivial knot in S 3 admits a lens space surgery with slope p, then p≤4g + 3, where g is the genus of the knot. This is a close approximation to a bound conjectured by Goda and Teragaito. 1. ..."
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Cited by 11 (0 self)
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Abstract. Using work of Ozsváth and Szabó, we show that if a nontrivial knot in S 3 admits a lens space surgery with slope p, then p≤4g + 3, where g is the genus of the knot. This is a close approximation to a bound conjectured by Goda and Teragaito. 1.
Capping off open books and the Ozsváth–Szabó contact invariant
"... Abstract. If (S, φ) is an open book with disconnected binding, then we can form a new open book (S′, φ′) by capping off one of the boundary components of S with a disk. LetMS,φ denote the 3manifold with open book decomposition (S, φ). We show that there is a Uequivariant map from HF+(−MS′,φ′) to H ..."
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Cited by 9 (2 self)
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Abstract. If (S, φ) is an open book with disconnected binding, then we can form a new open book (S′, φ′) by capping off one of the boundary components of S with a disk. LetMS,φ denote the 3manifold with open book decomposition (S, φ). We show that there is a Uequivariant map from HF+(−MS′,φ′) to HF+(−MS,φ) which sends c+(S′, φ′) to c+(S, φ), and we discuss various applications. In particular, we compute d3(ξ) for every contact manifold (M, ξ) for which c+(ξ) 6 = 0 and ξ is supported by a genus one open book with periodic monodromy. 1.
The Dehn surgery characterization of the trefoil and the figure eight knot
"... Abstract. We give a Dehn surgery characterization of the trefoil and the figure eight knots. These results are gotten by combining surgery formulas in Heegaard Floer homology from an earlier paper with the characterization of these knots in terms of their knot Floer homology given in a recent paper ..."
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Abstract. We give a Dehn surgery characterization of the trefoil and the figure eight knots. These results are gotten by combining surgery formulas in Heegaard Floer homology from an earlier paper with the characterization of these knots in terms of their knot Floer homology given in a recent paper of Ghiggini. 1.