A type D structure in Khovanov homology

Roberts, Lawrence P.

doi:10.1016/j.aim.2016.02.007

Cited by 12 publications

(46 citation statements)

References 10 publications

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“…The first theory is a reformulation of Khovanov's functor-valued invariant [1] in the bordered language. The second theory was introduced by Lawrence Roberts in [9] and [8].…”

Section: Introductionmentioning

confidence: 99%

“…Roberts [9] [8] has a different construction of a bordered theory for Khovanov homology, including a differential bigraded algebra BΓ n as well as Type D and Type A structures for tangles. The goal of Section 5 and Section 6 is to construct Roberts' theory using Khovanov's theory.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.0.5 (Proposition 6.2.7 and Proposition 6.3.10). The Type A structure A([T 2 ] Kh ) over BΓ n , and the Type D structure D([T 1 ] Kh ) over BΓ n , are isomorphic to the Type A and D structures Roberts associates to T 2 and T 1 in [8] and [9]. By Proposition 6.4.3, the pairing ⊠ is the same over B⊙m(B ! )…”

Section: Introductionmentioning

confidence: 99%

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On bordered theories for Khovanov homology

Manion¹

2017

Algebr. Geom. Topol.

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We describe how to formulate Khovanov's functor-valued invariant of tangles in the language of bordered Heegaard Floer homology. We then give an alternate construction of Lawrence Roberts' Type D and Type A structures in Khovanov homology, and his algebra $\mathcal{B}\Gamma_n$, in terms of Khovanov's theory of modules over the ring $H^n$. We reprove invariance and pairing properties of Roberts' bordered modules in this language. Along the way, we obtain an explicit generators-and-relations description of $H^n$ which may be of independent interest.Comment: 87 pages; 2 figure

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“…The first theory is a reformulation of Khovanov's functor-valued invariant [1] in the bordered language. The second theory was introduced by Lawrence Roberts in [9] and [8].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On bordered theories for Khovanov homology

Manion¹

2017

Algebr. Geom. Topol.

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“…Examples of analogous bordered theories developed for other invariants are: bordered Heegaard Floer homology [24], [25]; bordered theory for knot Floer homology [33], [34], [35]; bordered theories for Khovanov homology [36], [37], [27]. Step (3) for Heegaard Floer homology was done in [40], and for knot Floer homology in [34], [35].…”

mentioning

confidence: 99%

Bordered theory for pillowcase homology

Kotelskiy

2019

Mathematical Research Letters

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We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A∞ module M (L) over A. Then we prove that Lagrangian Floer homology HF (L, L ) is isomorphic to a suitable algebraic pairing of modules M (L) and M (L ). This extends the pillowcase homology construction given a 2-stranded tangle inside a 3-ball, if one obtains an immersed unobstructed Lagrangian inside the pillowcase, one can further associate an A∞ module to that Lagrangian.

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“…It would also be interesting to compare ours to other, more combinatorial constructions, of bordered Khovanov invariants, e.g. [29,30,27,16]. In light of the bordered interpretation, one would expect that the cohomology of the hom spaces between two 2-tangles is isomorphic to the Khovanov homology of their union.…”

Section: Introductionmentioning

confidence: 91%

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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A type D structure in Khovanov homology

Cited by 12 publications

References 10 publications

On bordered theories for Khovanov homology

On bordered theories for Khovanov homology

Bordered theory for pillowcase homology

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

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