On the equivalence of Legendrian and transverse invariants in knot Floer homology

Baldwin, John A.; Vela-Vick, David Shea; Vértesi, Vera

doi:10.2140/gt.2013.17.925

Cited by 34 publications

(49 citation statements)

References 35 publications

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“…Combined with our work in [4], it also shows that the 'LOSS' invariant in knot Floer homology satisfies a certain functoriality with respect to Lagrangian concordance. Unsurprisingly, our strategy for proving Theorem 1.9 relies on the combined work of Taubes [44][45][46][47][48] and Colin et al [6][7][8], which shows that there is an isomorphism…”

Section: The Morphism H Encodes Maps Hsupporting

confidence: 69%

“…Below, we describe some related results and several goals for future work. In our article [3], we prove that our contact invariant 'agrees' with Honda, Kazez, and Matić's E H invariant. Forgetting about naturality, we can view ψ(M, Γ, ξ ) as given by…”

Section: The Morphism H Encodes Maps Hsupporting

confidence: 58%

“…This point is illustrated in [4], where we use the contact invariant defined in this paper to construct new invariants of Legendrian and transverse knots in monopole knot homology. The functoriality of Kronheimer and Mrowka's invariant ψ under exact symplectic cobordism leads to a proof that our Legendrian invariant is functorial with respect to Lagrangian concordance (see [42] for a similar result), something which is not independently known to be true of the analogous 'LOSS' invariant in knot Floer homology [37] (see Section 1.3).…”

Section: Introductionmentioning

confidence: 99%

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A Contact Invariant in Sutured Monopole Homology

Baldwin

Sivek

2016

Forum of Mathematics, Sigma

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We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, and Matić's contact invariant in sutured Heegaard Floer homology (SFH). In the process of defining our invariant, we construct maps on SHM associated to contact handle attachments, analogous to those defined by Honda, Kazez, and Matić in SFH. We use these maps to establish a bypass exact triangle in SHM analogous to Honda's in SFH. This paper also provides the topological basis for the construction of similar gluing maps in sutured instanton Floer homology, which are used in Baldwin and Sivek [Selecta Math. (N.S.), 22(2) (2016), 939-978] to define a contact invariant in the instanton Floer setting.

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Section: The Morphism H Encodes Maps Hsupporting

confidence: 69%

Section: The Morphism H Encodes Maps Hsupporting

confidence: 58%

Section: Introductionmentioning

confidence: 99%

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A Contact Invariant in Sutured Monopole Homology

Baldwin

Sivek

2016

Forum of Mathematics, Sigma

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“…Each basepoint in a multi-pointed Heegaard diagram for (L, p) determines a differential on HFK(L, p). These basepoint actions have previously been applied in [BL12,BVVV13,Sar15,BLS,Zem16]. The combined actions, subject to anticommutation relations described in Subsection 3.3, make HFK(L, p) a Clifford module over a Clifford algebra Ω ν .…”

Section: T1mentioning

confidence: 99%

On Conway mutation and link homology

Lambert-Cole

2019

Advances in Mathematics

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We give a new, elementary proof that Khovanov homology with Z/2Z-coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that δ-graded knot Floer homology is mutation-invariant. Using the Clifford module structure on HFK induced by basepoint maps, we carry out this strategy for mutations on a large class of tangles. Let L ′ be a link obtained from L by mutating the tangle T . Suppose some rational closure of T corresponding to the mutation is the unlink on any number of components. Then L and L ′ have isomorphic δ-graded HFK groups over Z/2Z as well as isomorphic Khovanov homology over Q. We apply these results to establish mutation-invariance for the infinite families of Kinoshita-Terasaka and Conway knots. Finally, we give sufficient conditions for a general Khovanov-Floer theory to be mutation-invariant.2010 Mathematics Subject Classification. 57M27; 57R58.The complex (CKh(D), d Kh ) possess two gradings, the quantum and homological grading, and the differential preserves the quantum grading and increases the homological grading by one. Thus the homology Kh(D) splits into the direct sum of bigraded modules Kh i,j (D) where i denotes the homological grading and j the quantum grading.Let p be a fixed basepoint in the plane contained in the diagram D. The basepoint p determines a chain map X p : CKh(D) → CKh(D) that squares to 0. The kernel of X p is a subcomplex and reduced Khovanov homology is its homology, with a shift in the quantum grading:It is an invariant of L up to isotopies supported away from p. While over Z the reduced homology depends on the component containing the basepoint, this is not true over Z/2Z.for every link L and any basepoint p. In particular, Kh(L) is well-defined independent of p.Consequently, when discussing reduced Khovanov homology over Z/2Z we will suppress any mention of the basepoint p.Khovanov homology satisfies Kunneth-type formulas for disjoint unions and connected sums. The following properties are well-known and the proofs are easy deductions from the definition of Khovanov homology and Proposition 2.1.Lemma 2.2. Let L 1 , L 2 be arbitrary links. Over Z/2Z there are isomorphismsfor any choice of connected sum.Khovanov homology satisfies an unoriented skein exact triangle. Fix a crossing of D and let D 0 , D 1 denote the 0-and 1-resolutions of D at this crossing. Since Khovanov homology is computed from a cube of resolutions, the complex CKh(D) is, up to a grading shift, the mapping cone of a chain map

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“…As a natural counterpart of right-veering mapping classes, right-veering closed braids (with respect to general open books) have been defined and studied in the literature [2,3,28]. In Section 2 we define a closed braid L in open book (S, φ) and discuss how to assign an element [ϕ L ] of the mapping class group for L. Then as a counterpart of the FDTC c(φ, C), we define c(φ, L, C) the FDTC for a closed braid L with respect to an open book (S, φ) and a boundary component C of S. The definitions given in this paper are more rigorous than those in [20].…”

Section: Introductionmentioning

confidence: 99%

Quasi-right-veering braids and nonloose links

Ito

Kawamuro

2019

Algebr. Geom. Topol.

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We introduce a notion of "quasi-right-veering" for closed braids, which plays an analogous role to "right-veering" for open books. We show that a transverse link K in a contact 3-manifold (M, ξ) is non-loose if and only if every braid representative of K with respect to every open book decomposition that supports (M, ξ) is quasi-right-veering. We also show that several definitions of "right-veering" closed braids are equivalent.

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On the equivalence of Legendrian and transverse invariants in knot Floer homology

Cited by 34 publications

References 35 publications

A Contact Invariant in Sutured Monopole Homology

A Contact Invariant in Sutured Monopole Homology

On Conway mutation and link homology

Quasi-right-veering braids and nonloose links

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