Symmetric Khovanov-Rozansky link homologies

Robert, Louis-Hadrien; Wagner, Emmanuel

doi:10.5802/jep.124

Symmetric Khovanov-Rozansky link homologies
[Homologies d’entrelacs de Khovanov–Rozansky symétriques]

Robert, Louis-Hadrien ¹ ; Wagner, Emmanuel ²

¹ Université de Genève 2–4 rue du lièvre, 1227 Genève, Switzerland Cogitamus Laboratory
² Univ Paris Diderot, IMJ-PRG, UMR 7586 CNRS F-75013, Paris, France Université de Bourgogne Franche-Comté, IMB, UMR 5584 21000 Dijon, France Cogitamus Laboratory

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 573-651.

Résumé
Abstract

On donne une catégorification de l’évaluation symétrique des toiles ${𝔰𝔩}_{N}$ en utilisant les mousses. On en déduit des théories homologiques d’entrelacs qui catégorifient les invariants quantiques d’entrelacs associés aux puissances symétriques de la représentation standard de ${𝔰𝔩}_{N}$ . Ces théories sont obtenues dans un cadre équivariant. On montre qu’il existe des suites spectrales de l’homologie triplement graduée de Khovanov-Rozansky vers ces homologies symétriques. On donne aussi une interpretation des bimodules de Soergel en terme de mousses.

We provide a finite-dimensional categorification of the symmetric evaluation of ${𝔰𝔩}_{N}$ -webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of ${𝔰𝔩}_{N}$ . The construction is made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov-Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.

Reçu le : 2018-07-13
Accepté le : 2020-03-16
Publié le : 2020-04-02

DOI : 10.5802/jep.124

Classification : 57R56, 57M27, 17B10, 17B37
Keywords: Link homology, quantum invariant, foams, Soergel bimodules
Mot clés : Homologie d’entrelacs, invariant quantiques, mousses, bimodules de Soergel

Affiliations des auteurs :

Robert, Louis-Hadrien ¹ ; Wagner, Emmanuel ²

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     author = {Robert, Louis-Hadrien and Wagner, Emmanuel},
     title = {Symmetric {Khovanov-Rozansky} link homologies},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {573--651},
     publisher = {Ecole polytechnique},
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     doi = {10.5802/jep.124},
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     url = {http://www.numdam.org/articles/10.5802/jep.124/}
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Robert, Louis-Hadrien; Wagner, Emmanuel. Symmetric Khovanov-Rozansky link homologies. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 573-651. doi : 10.5802/jep.124. http://www.numdam.org/articles/10.5802/jep.124/

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