Symmetric Khovanov-Rozansky link homologies

Louis-Hadrien Robert; Emmanuel Wagner

doi:10.5802/jep.124

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

Louis-Hadrien Robert ; Emmanuel Wagner

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 : https://doi.org/10.5802/jep.124
Classification : 57R56, 57M27, 17B10, 17B37
Mots clés : Homologie d’entrelacs, invariant quantiques, mousses, bimodules de Soergel

@article{JEP_2020__7__573_0,
     author = {Louis-Hadrien Robert and Emmanuel Wagner},
     title = {Symmetric {Khovanov-Rozansky} link homologies},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {573--651},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.124},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.124/}
}

Louis-Hadrien Robert; Emmanuel Wagner. Symmetric Khovanov-Rozansky link homologies. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020) , pp. 573-651. doi : 10.5802/jep.124. https://jep.centre-mersenne.org/articles/10.5802/jep.124/

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