Functoriality of colored link homologies

Ehrig, Michael; Tubbenhauer, Daniel; Wedrich, Paul

doi:10.1112/plms.12154

Cited by 28 publications

(50 citation statements)

References 47 publications

(123 reference statements)

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“…(c) Foams are suitably decorated 2-dimensional CW-complexes, defined abstractly or embedded in R 3 . They originate and most prominently appear in the study of link homologies, see for example [Kho04], [EST17], [RW20] or [ETW18]. Using the universal construction from [BHMV95], they can easily modified, see e.g.…”

Section: Symbolmentioning

confidence: 99%

Monoidal categories, representation gap and cryptography

Khovanov¹,

Sitaraman²,

Tubbenhauer³

2022

Preprint

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The linear decomposition attack provides a serious obstacle to direct applications of noncommutative groups and monoids in cryptography. To overcome this issue we propose to look at monoids with only big representations, in the sense made precise in the paper, and undertake a systematic study of such monoids. One of our main tools is Green's theory of cells (Green's relations).A large supply of monoids is delivered by monoidal categories. We consider simple examples of monoidal categories of diagrammatic origin, including the Temperley-Lieb, the Brauer and partition categories, and discuss lower bounds for their representations.

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Section: Symbolmentioning

confidence: 99%

Monoidal categories, representation gap and cryptography

Khovanov¹,

Sitaraman²,

Tubbenhauer³

2022

Preprint

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“…We expect that a proper generalization of this paper would prove equivalence of both (bi)categories, hence, also of the associated link homologies. Notice that functoriality of gl N homology has been shown in [15] using enhanced foams.…”

Section: Further Generalizationsmentioning

confidence: 99%

On the functoriality of sl(2) tangle homology

Beliakova¹,

Hogancamp²,

Putyra³

et al. 2019

Preprint

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We construct an explicit equivalence between the (bi)category of gl 2 webs and foams and the Bar-Natan (bi)category of Temperley-Lieb diagrams and cobordisms. With this equivalence we can fix functoriality of every link homology theory that factors through the Bar-Natan category. To achieve this, we define web versions of arc algebras and their quasi-hereditary covers, which provide strictly functorial tangle homologies. Furthermore, we construct explicit isomorphisms between these algebras and the original ones based on Temperley-Lieb cup diagrams. e immediate application is a strictly functorial version of the Beliakova-Putyra-Wehrli quantization of the annular link homology.

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“…1. These are indeed chain complexes, since when one forgets the H ′ -module structure, they are precisely the maps used to define the classical Rickard complexes, see for instance [ETW18].…”

Section: Rickard Complexesmentioning

confidence: 99%

A categorification of the colored Jones polynomial at a root of unity

Qi,

Robert,

Sussan

et al. 2021

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There is a p-differential on the triply-graded Khovanov-Rozansky homology of knots and links over a field of positive characteristic p that gives rise to an invariant in the homotopy category finite-dimensional p-complexes.A differential on triply-graded homology discovered by Cautis is compatible with the p-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a 2pth root of unity.

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Functoriality of colored link homologies

Abstract: We prove that the bigraded, colored Khovanov–Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.

Cited by 28 publications

References 47 publications

Monoidal categories, representation gap and cryptography

Monoidal categories, representation gap and cryptography

On the functoriality of sl(2) tangle homology

A categorification of the colored Jones polynomial at a root of unity

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