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118
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 68 (9 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 65 (1 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.
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 43 (3 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.
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 33 (3 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.
Knot Floer homology and rational surgeries
"... Abstract. Let K be a rationally nullhomologous knot in a threemanifold Y. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a threemanifold obtained as Morse surgery on the knot K. As an application, we express the Heegaard Floer homol ..."
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Cited by 28 (2 self)
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Abstract. Let K be a rationally nullhomologous knot in a threemanifold Y. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a threemanifold obtained as Morse surgery on the knot K. As an application, we express the Heegaard Floer homology of rational surgeries on Y along a nullhomologous knot K in terms of the filtered homotopy type of the knot invariant for K. This has applications to Dehn surgery problems for knots in S 3. In a different direction, we use the techniques developed here to calculate the Heegaard Floer homology of an arbitrary Seifert fibered threemanifold. 1.
The OzsváthSzabó and Rasmussen concordance invariants are not equal
, 2005
"... 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 24 (3 self)
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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.
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 22 (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.
Khovanov homology is an unknotdetector
"... Abstract. We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then s ..."
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Cited by 22 (2 self)
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Abstract. We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot. 1
A link surgeries spectral sequence for monopole Floer homology
"... Abstract. To a link L ⊂ S 3, we associate a spectral sequence whose E 2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E k for k ≥ 2 depend only on the mutation equivalence class of L. We define a mod ..."
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Cited by 22 (3 self)
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Abstract. To a link L ⊂ S 3, we associate a spectral sequence whose E 2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E k for k ≥ 2 depend only on the mutation equivalence class of L. We define a mod 2 grading on the spectral sequence which interpolates between the δgrading on Khovanov homology and the mod 2 grading on monopole Floer homology. More generally, we construct new invariants of a framed link in a 3manifold as the pages of a spectral sequence modeled on the surgery exact triangle. The differentials count SeibergWitten monopoles over families of metrics parameterized by permutohedra. We make extensive use of a surprising connection between the topology of link surgeries and the combinatorics of graph associahedra. This connection also yields remarkably simple realizations of the permutohedra and associahedra, as refinements of the hypercube. Contents