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33
Twisted Alexander polynomials detect fibered 3manifolds
 Monopoles and ThreeManifolds, New Mathematical Monographs (No. 10), Cambridge University Press. , Knots, sutures and excision
"... Abstract. A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3–manifolds. In this paper we show that ..."
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Cited by 9 (1 self)
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Abstract. A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3–manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3–manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S 1 × N 3 admits a symplectic structure, then N fibers over S 1. In fact we will completely determine the symplectic cone of S 1 × N in terms of the fibered faces of the Thurston norm ball of N. 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 7 (2 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
Knots, sutures and excision
"... Abstract. We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of [12]. Applications include a new proof of Property P for knots. Contents ..."
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Cited by 5 (0 self)
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Abstract. We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of [12]. Applications include a new proof of Property P for knots. Contents
Exotic group actions on simply connected smooth 4manifolds
, 2009
"... We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4manifolds with nontrivial SeibergWitten invariants. ..."
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Cited by 5 (0 self)
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We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4manifolds with nontrivial SeibergWitten invariants.
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 4 (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
The embedded contact homology index revisited
"... Let Y be a closed oriented 3manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This in ..."
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Cited by 3 (2 self)
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Let Y be a closed oriented 3manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the relative grading on ECH; the ECH differential counts holomorphic curves in the symplectization of Y whose relative homology classes have ECH index 1. A known index inequality implies that such curves are (mostly) embedded and satisfy some additional constraints. In this paper we prove four new results about the ECH index. First, we refine the relative grading on ECH to an absolute grading, which associates to each union of Reeb orbits a homotopy class of oriented 2plane fields on Y. Second, we extend the ECH index inequality to symplectic cobordisms between threemanifolds with Hamiltonian structures, and simplify the proof. Third, we establish general inequalities on the ECH index of unions and multiple covers of holomorphic curves in cobordisms. Finally, we define a new relative filtration on ECH, or any other kind of contact homology of a contact 3manifold, which is similar to the ECH index and related to the Euler characteristic of holomorphic curves. This does not give new topological invariants except possibly in special situations, but it is a useful computational tool.
Instanton Floer homology and the Alexander polynomial
"... Abstract. The instanton Floer homology of a knot in S 3 is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree, arising from the 2dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the gener ..."
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Cited by 3 (2 self)
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Abstract. The instanton Floer homology of a knot in S 3 is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree, arising from the 2dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces of this endomorphism. We show that the Euler characteristics of these generalized eigenspaces are the coefficients of the Alexander polynomial of the knot. Among other applications, we deduce that instanton homology detects fibered knots. 1
Periodic Floer homology and SeibergWitten Floer cohomology
, 2009
"... Various SeibergWitten Floer cohomologies are defined for a closed, oriented 3manifold; and if it is the mapping torus of an areapreserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of S ..."
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Cited by 3 (0 self)
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Various SeibergWitten Floer cohomologies are defined for a closed, oriented 3manifold; and if it is the mapping torus of an areapreserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of SeibergWitten Floer cohomology and the corresponding periodic Floer homology, and describe some immediate consequences.
Addendum to: “Knots, sutures and excision
"... We observe that the main theorem in [4] immediately implies its analogue for closed 3–manifolds. Theorem 1. Suppose Y is a closed irreducible 3–manifold, F ⊂ Y is a closed connected surface of genus g ≥ 2. If HM•(Y F) ∼ = Z, then Y fibers over the circle with F as a fiber. The case that g = 1 is ..."
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Cited by 2 (0 self)
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We observe that the main theorem in [4] immediately implies its analogue for closed 3–manifolds. Theorem 1. Suppose Y is a closed irreducible 3–manifold, F ⊂ Y is a closed connected surface of genus g ≥ 2. If HM•(Y F) ∼ = Z, then Y fibers over the circle with F as a fiber. The case that g = 1 is already treated in [3], following the argument of Ghiggini [1]. Theorem 2. [3, Theorem 42.7.1] Suppose Y is a closed irreducible 3–manifold, F ⊂ Y is a torus, η is a 1–cycle in Y that intersects F once. If HM•(Y F, Γη) ∼ = R, then Y fibers over the circle with F as a fiber. Remark 3. The statement of [3, Theorem 42.7.1] uses a field Kη of characteristic 2, because the proof involves the surgery exact sequence whose proof requires characteristic 2. Kronheimer pointed out that this part can be replaced by the Excision Theorem [4, Theorem 3.2], which allows us to use any characteristic.