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58
Monopoles and lens space surgeries
 ArXive:math.GT/0310164
, 2003
"... Abstract. Monopole Floer homology is used to prove that real projective threespace cannot be obtained from Dehn surgery on a nontrivial knot in the threesphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which sh ..."
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Cited by 34 (10 self)
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Abstract. Monopole Floer homology is used to prove that real projective threespace cannot be obtained from Dehn surgery on a nontrivial knot in the threesphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of threemanifolds which do not admit taut foliations. 1.
ON THE KHOVANOV AND KNOT FLOER HOMOLOGIES OF QUASIALTERNATING LINKS
, 2008
"... Quasialternating links are a natural generalization of alternating links. In this paper, we show that quasialternating links are “homologically thin ” for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, to ..."
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Cited by 23 (0 self)
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Quasialternating links are a natural generalization of alternating links. In this paper, we show that quasialternating links are “homologically thin ” for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing.
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 18 (2 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.
A concordance invariant from the Floer homology of double branched covers
, 2005
"... Ozsváth and Szabó defined an analog of the Frøyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3sphere branched over a knot K, we obtain an invariant δ of knot concordance. We show that δ is determined by the sign ..."
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Cited by 16 (0 self)
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Ozsváth and Szabó defined an analog of the Frøyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3sphere branched over a knot K, we obtain an invariant δ of knot concordance. We show that δ is determined by the signature for alternating knots and knots with up to nine crossings, and conjecture a similar relation for all Hthin knots. We also use δ to prove that for all knots K with τ(K)> 0, the positive untwisted double of K is not smoothly slice.
The OzsváthSzabó and Rasmussen concordance invariants are not equal
, 2005
"... Abstract. In this paper we present several counterexamples to Rasmussen’s conjecture that the concordance invariant coming from Khovanov homology is equal to twice the invariant coming from OzsváthSzabó Floer homology. The counterexamples are twisted Whitehead doubles of the (2, 2n + 1) torus knots ..."
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Cited by 14 (2 self)
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Abstract. In this paper we present several counterexamples to Rasmussen’s conjecture that the concordance invariant coming from Khovanov homology is equal to twice the invariant coming from OzsváthSzabó Floer homology. The counterexamples are twisted Whitehead doubles of the (2, 2n + 1) torus knots. 1.
KNOT FLOER HOMOLOGY AND INTEGER SURGERIES
, 2007
"... Abstract. Let Y be a closed threemanifold with trivial first homology, and let K ⊂ Y be a knot. We give a description of the Heegaard Floer homology of integer surgeries on Y along K in terms of the filtered homotopy type of the knot invariant for K. As an illustration of these techniques, we calcu ..."
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Cited by 11 (2 self)
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Abstract. Let Y be a closed threemanifold with trivial first homology, and let K ⊂ Y be a knot. We give a description of the Heegaard Floer homology of integer surgeries on Y along K in terms of the filtered homotopy type of the knot invariant for K. As an illustration of these techniques, we calculate the Heegaard Floer homology groups of nontrivial circle bundles over Riemann surfaces (with coefficients in Z/2Z). 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 10 (1 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.
Knot polynomials and knot homologies
"... Abstract. This is an expository paper discussing some parallels between the Khovanov and knot Floer homologies. We describe the formal similarities between the theories, and give some examples which illustrate a somewhat mysterious correspondence between them. 1 ..."
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Cited by 9 (2 self)
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Abstract. This is an expository paper discussing some parallels between the Khovanov and knot Floer homologies. We describe the formal similarities between the theories, and give some examples which illustrate a somewhat mysterious correspondence between them. 1
Heegaard Floer homology and genus one, one boundary component open books. arXiv:0804.3624
, 2008
"... Abstract. 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 T 2bundle over S 1 with first Betti number ..."
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Cited by 7 (2 self)
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Abstract. 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 T 2bundle over S 1 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. 1.