Bordered theory for pillowcase homology

Kotelskiy, Artem

doi:10.4310/mrl.2019.v26.n5.a11

Cited by 4 publications

(2 citation statements)

References 57 publications

(115 reference statements)

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“…In words, the Khovanov homology of the link arising as the union of a tangle T with the mirror of a tangle V can be recovered, up to tensoring with a particular two dimensional vector space, as the cohomology of the space of morphisms between the twisted complexes we associate to V and T , respectively. We should point out that similar results in these directions, and more precise comparisons with bordered Floer homology, have been obtained independently by Kotelskiy [21,20].…”

Section: Introductionsupporting

confidence: 81%

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology

Hedden

Herald

Hogancamp

et al. 2020

Trans. Amer. Math. Soc.

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For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless SU (2) character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka's singular instanton link homology.

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Section: Introductionsupporting

confidence: 81%

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology

Hedden

Herald

Hogancamp

et al. 2020

Trans. Amer. Math. Soc.

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“…In the context of instanton knot Floer homology, for 4-ended tangles, this leads to representation theoretic constructions of immersed curves R π (T ) and R π (T ) on the pillowcase due to Hedden, Herald and Kirk [HHK14,HHK18]. The first author extended this construction algebraically [Kot19]. In Heegaard Floer homology, Auroux [Aur10a,Aur10b] and Lekili and Perutz [LP11] suggested that the correct moduli space for the invariant of a compact oriented 3-manifold M with ∂M = Σ g and a basepoint z ∈ Σ g should be Sym g (Σ g z).…”

Section: T (∂I ∂I)mentioning

confidence: 99%

Immersed curves in Khovanov homology

Kotelskiy¹,

Watson²,

Zibrowius³

2019

Preprint

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We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles T multicurves BN(T ), that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use BN(T ) to define two immersed curve invariants Kh(T ) and Kh(T ), which satisfy similar gluing theorems that recover reduced and unreduced Khovanov homology of links, respectively. As a first application, we prove that Conway mutation preserves reduced Bar-Natan homology over the field with two elements and Rasmussen's s-invariant over any field. As a second application, we give a geometric interpretation of Rozansky's categorification of the two-stranded Jones-Wenzl projector. This allows us to define a module structure on reduced Bar-Natan and Khovanov homologies of infinitely twisted knots, generalizing a result by Benheddi. Contents 1. Introduction 1 2. Algebraic preliminaries 11 3. Khovanov-theoretic invariants of links 17 4. Bar-Natan's universal cobordism category and tangle invariants 25 5. Classification results 36 6. Immersed curve invariants 56 7. Pairing theorem 61 8. The mapping class group action 71 9. Signs and mutation 79 10. Khovanov homology of infinitely twisted knots 84 References 93 AK is supported by an AMS-Simons travel grant. LW is supported by an NSERC discovery/accelerator grant and was partially supported by funding from the Simons Foundation and the Centre de Recherches Mathématiques, through the Simons-CRM scholar-in-residence program.

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Bordered Floer homology for manifolds with torus boundary via immersed curves

Hanselman¹,

Rasmussen²,

Watson³

2023

J. Amer. Math. Soc.

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This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M M is such a manifold, we show that the type D structure C F D ^ ( M ) \widehat {CFD}(M) may be viewed as a set of immersed curves decorated with local systems in ∂ M \partial M . These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of H F ^ \widehat {HF} decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of H F ^ \widehat {HF} . In particular, it follows that a prime rational homology sphere Y Y with H F ^ ( Y ) > 5 \widehat {HF}(Y)>5 must be geometric. Other results include a new proof of Eftekhary’s theorem that L-space homology spheres are atoroidal; a complete characterization of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.

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Bordered theory for pillowcase homology

Cited by 4 publications

References 57 publications

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology

Immersed curves in Khovanov homology

Bordered Floer homology for manifolds with torus boundary via immersed curves

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