Sutured Floer homology and invariants of Legendrian and transverse knots

Etnyre, John B.; Vela-Vick, David Shea; Zarev, Rumen

doi:10.2140/gt.2017.21.1469

Cited by 26 publications

(35 citation statements)

References 41 publications

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“…The resulting 4-manifold is exactly W 2 . Hence, from Proposition 2.5 in [15], (7) holds true and we conclude the proof of lemma 3.10. Proof.…”

Section: A Reformulation Of Canonical Mapssupporting

confidence: 69%

“…The commutativity of ( 1) is guaranteed by the functoriality of the gluing map. The crucial difference from the work of Etnyre, Vela-Vick and Zarev in [7] is that, because of the involvement of closures, the construction of the grading in the monopole and the instanton settings is a delicate issue. We construct a grading in the direct limit in two steps.…”

Section: Outline Of the Proofmentioning

confidence: 93%

“…The construction of the (canonical) module KHM ´pY, K, pq was inspired by Etnyre, Vela-Vick and Zarev in [7], where they use a sequence of balanced sutured manifolds pY pKq, Γ n q and the gluing maps in sutured (Heegaard) Floer theory, which was introduced by Honda, Kazez, and Matić [12], to construct a direct system. They proved that the direct limit is isomorphic to the classical minus version of knot Floer homology in Heegaard Floer theory.…”

Section: Outline Of the Proofmentioning

confidence: 99%

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Knot homologies in monopole and instanton theories via sutures

Li¹

2019

Preprint

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In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot K Ă Y and a base point p P K, we can associate the minus versions, KHM ´pY, K, pq and KHI ´pY, K, pq, to the triple pY, K, pq. We prove that a Seifert surface of K induces a Z-grading, and there is an U -map on the minus versions, which is of degree ´1. We also prove other basic properties of them. If K Ă Y is not null-homologous but represents a torsion class, then we can also construct the corresponding minus versions for pY, K, pq. We also proved a surgery-type formula relating the minus versions of a knot K with those of the dual knot, when performing a Dehn surgery of large enough slope along K. The techniques developed in this paper can also be applied to compute the sutured monopole and instanton Floer homologies of any sutured solid tori.

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“…The resulting 4-manifold is exactly W 2 . Hence, from Proposition 2.5 in [15], (7) holds true and we conclude the proof of lemma 3.10. Proof.…”

Section: A Reformulation Of Canonical Mapssupporting

confidence: 69%

Section: Outline Of the Proofmentioning

confidence: 93%

Section: Outline Of the Proofmentioning

confidence: 99%

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Knot homologies in monopole and instanton theories via sutures

Li¹

2019

Preprint

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“…We will do this in future work. Once we do, we will be able to define a "minus" (or "from") version of KHM by following a scheme of Etnyre, Vela-Vick and Zarev for recovering the "minus" version of knot Floer homology from SF H using bypass attachment maps [7]. In the meantime, we use these handle attachment maps in [1] to prove a monopole Floer analogue of Honda's bypass exact triangle in SF H.…”

Section: Introductionmentioning

confidence: 99%

Naturality in sutured monopole and instanton homology

Baldwin¹,

Sivek²

2015

J. Differential Geom.

160

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Abstract. In [16], Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce refinements of these invariants which assign much richer algebraic objects called projectively transitive systems of modules to balanced sutured manifolds and isomorphisms of such systems to diffeomorphisms of balanced sutured manifolds. Our work provides the foundation for extending these sutured Floer theories to other interesting functorial frameworks as well, and can be used to construct new invariants of contact structures and (perhaps) of knots and bordered 3-manifolds.

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“…Let K be a framed null-homologous knot in Y . The first author [14] and Etnyre, Vela-Vick, and Zarev [12] showed that the limit of SFH (M, γ n ) along the maps σ + is HFK − (−Y, K). Furthermore, if K is Legendrian, then EH (K) limits to the LOSS invariant L − (K).…”

mentioning

confidence: 99%