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56
Floer homology and knot complements
, 2003
"... Abstract. We use the OzsváthSzabó theory of Floer homology to define an invariant of knot complements in threemanifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the OzsváthSzabó Floer homology of large integral surgeries on the k ..."
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Cited by 238 (7 self)
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Abstract. We use the OzsváthSzabó theory of Floer homology to define an invariant of knot complements in threemanifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the OzsváthSzabó Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of perfect knots in S3 for which ĈF r has a particularly simple form. For these knots, formal properties of the OzsváthSzabó theory enable us to make a complete calculation of the Floer homology. It turns out that most small knots are perfect. 1.
HOLOMORPHIC DISKS AND THREEMANIFOLD INVARIANTS: PROPERTIES AND APPLICATIONS
, 2001
"... ... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. The properties include a ..."
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Cited by 195 (29 self)
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... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. The properties include a relationship between the Euler characteristics of HF ± and Turaev’s torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to threemanifold topology.
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.
The Symplectic Thom Conjecture
"... In this paper, we demonstrate a relation among SeibergWitten invariants which arises from embedded surfaces in fourmanifolds whose selfintersection number is negative. These relations, together with Taubes' basic theorems on the SeibergWitten invariants of symplectic manifolds, are then use ..."
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Cited by 56 (8 self)
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In this paper, we demonstrate a relation among SeibergWitten invariants which arises from embedded surfaces in fourmanifolds whose selfintersection number is negative. These relations, together with Taubes' basic theorems on the SeibergWitten invariants of symplectic manifolds, are then used to prove the Symplectic Thom Conjecture: a symplectic surface in a symplectic fourmanifold is genusminimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative selfintersection in fourmanifolds. 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 32 (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.
SeibergWitten invariants of 4manifolds with free circle actions
 Commun. Contemp. Math
, 2001
"... The main result of this paper describes a formula for the SeibergWitten invariant of a 4manifold X which admits a nontrivial free S 1action. A free circle action on X is classified by its orbit space, a 3manifold M, and its Euler class χ ∈ H 2 (M; Z). If χ = 0, then ..."
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Cited by 26 (4 self)
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The main result of this paper describes a formula for the SeibergWitten invariant of a 4manifold X which admits a nontrivial free S 1action. A free circle action on X is classified by its orbit space, a 3manifold M, and its Euler class χ ∈ H 2 (M; Z). If χ = 0, then
Seifert fibered contact threemanifolds via surgery
, 2003
"... Using contact surgery we define families of contact structures on certain Seifert fibered three–manifolds. We prove that all these contact structures are tight using Ozsváth–Szabó’s contact invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered thre ..."
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Cited by 25 (8 self)
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Using contact surgery we define families of contact structures on certain Seifert fibered three–manifolds. We prove that all these contact structures are tight using Ozsváth–Szabó’s contact invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three–manifold carrying at least n pairwise non– isomorphic tight, not semi–fillable contact structures.
Higher Type Adjunction Inequalities In SeibergWitten Theory
, 1998
"... In this paper, we derive new adjunction inequalities for embedded surfaces with nonnegative selfintersection number in fourmanifolds. These formulas are proved by using relations between SeibergWitten invariants which are induced from embedded surfaces. To prove these relations, we develop the r ..."
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Cited by 24 (5 self)
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In this paper, we derive new adjunction inequalities for embedded surfaces with nonnegative selfintersection number in fourmanifolds. These formulas are proved by using relations between SeibergWitten invariants which are induced from embedded surfaces. To prove these relations, we develop the relevant parts of a Floer theory for fourmanifolds which bound circlebundles over Riemann surfaces. 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