Fixing the functoriality of Khovanov homology: A simple approach

Sano, Taketo

doi:10.1142/s0218216521500747

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…where P .F; c/ and Q.F; c/ are as defined in (36) and (37). In other words, a type 1 point p on an i-colored facet contributes a factor of p x j C x k to the evaluation hF; ci ı .…”

Section: Unlabeled Anchor Points and Bigon Decompositionmentioning

confidence: 99%

“…Working with that larger ring and U.1/ N -equivariant cohomology is a recent phenomenon. It was used by T Sano [37] in resolving the minus sign ambiguity in the functorial extension of Khovanov homology to link cobordisms, bypassing earlier constructions that required additional decorations of links and cobordisms (see [19] for more references and a short discussion). We expect this symmetry breaking of the ground ring generators to find more applications to link homology in the future.…”

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confidence: 99%

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Anchored foams and annular homology

Akhmechet,

Khovanov

2023

Algebr. Geom. Topol.

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We describe equivariant SL.2/ and SL.3/ homology for links in the thickened annulus via foam evaluation. The thickened annulus is replaced by 3-space with a distinguished line in it. Generators of state spaces for annular webs are represented by foams with boundary that may intersect the distinguished line; intersection points, called anchor points, contribute additional terms, reminiscent of square roots of the Hessian, to the foam evaluation. Both oriented and unoriented SL.3/ foams are treated. 57K18; 18N25, 57K16 1 IntroductionAsaeda-Przytycki-Sikora [2] homology of links in the thickened annulus has led to a number of interesting developments -see the first author [1], Baldwin, Beliakova, Grigsby, Licata, Putyra and Wehrli [3;5;11;12;13] and Roberts [35] -and extensions of their work to SL.N / and GL.N / link homology in the thickened annulus -see Queffelec, Rose, Sartori and Wedrich [30;31;32]. GL.N / and SL.N / link homology theories are closely related to foam evaluation. This connection was made the most transparent by the work of Robert and Wagner [34], who wrote down a combinatorial formula for GL.N / closed foam evaluation that allows to build GL.N / link homology from the ground up, bypassing categorical approaches to the latter. A variation of their formula was used by Robert and the second author [18] to evaluate unoriented SL.3/ foams, giving a combinatorial approach to some of the structures discovered by Kronheimer and Mrowka [23].In this paper we extend foam evaluation framework to build equivariant SL.2/ and SL.3/ state spaces for annular webs and, consequently, equivariant SL.2/ and SL.3/ homology for links in the thickened annulus. Our construction complements earlier work [30; 32] on the subject. The same approach allows us to define state spaces for

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“…where P .F; c/ and Q.F; c/ are as defined in (36) and (37). In other words, a type 1 point p on an i-colored facet contributes a factor of p x j C x k to the evaluation hF; ci ı .…”

Section: Unlabeled Anchor Points and Bigon Decompositionmentioning

confidence: 99%

mentioning

confidence: 99%

Anchored foams and annular homology

Akhmechet,

Khovanov

2023

Algebr. Geom. Topol.

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“…Furthermore, it lifts to a projective functor (well-defined on 2-morphisms up to an overall sign) from the 2-category of tangle cobordisms to the 2-category of complexes of bimodules over H n , over all n ≥ 0, and maps of complexes, up to homotopy [16,89] (see also [72] for another proof). Taking care of the sign is subtle; see [23,31,35,152].…”

Section: Categorification Of the Jones Polynomial For Links And Tanglesmentioning

confidence: 99%

Categorical lifting of the Jones polynomial: a survey

Khovanov

Lipshitz

2022

Bull. Amer. Math. Soc.

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“…Both of these approaches require expanding the coefficient ring to include

i

(the primitive fourth root of unity) and keeping track of more topological data than just oriented cobordisms. (A recent preprint by Sano [24] provides an alternate approach that requires adjusting the signs of the maps.) Because the injectivity statement of Theorem 1.4 does not require pinning down the sign, we opted to stick with the simpler framework from [3], at the cost of maintaining the sign indeterminacy.…”

Section: Introductionmentioning

confidence: 99%

Khovanov homology and cobordisms between split links

Gujral

Levine

2022

Journal of Topology

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In this paper, we study the (in)sensitivity of the Khovanov functor to 4‐dimensional linking of surfaces. We prove that if L$L$ and L′$L^{\prime }$ are split links, and C$C$ is a cobordism between L$L$ and L′$L^{\prime }$ that is the union of disjoint (but possibly linked) cobordisms between the components of L$L$ and the components of L′$L^{\prime }$, then the map on Khovanov homology induced by C$C$ is completely determined by the maps induced by the individual components of C$C$ and does not detect the linking between the components. As a corollary, we prove that a strongly homotopy–ribbon concordance (that is, a concordance whose complement can be built with only 1‐ and 2‐handles) induces an injection on Khovanov homology, which generalizes a result of the second author and Zemke. Additionally, we show that a non‐split link cannot be ribbon concordant to a split link.

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Fixing the functoriality of Khovanov homology: A simple approach

Cited by 6 publications

References 17 publications

Anchored foams and annular homology

Anchored foams and annular homology

Categorical lifting of the Jones polynomial: a survey

Khovanov homology and cobordisms between split links

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