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.
Highlights
In [RW17], we provided a combinatorial evaluation of the foams underlying the colored Khovanov-Rozansky link homologies [CK08a, CK08b, MS09, Sus07, MSV09, Wu14, Yon11]
The present paper grew up as an attempt to provide a similar formula for foams underlying link homologies categorifying the Reshetikhin-Turaev invariants of links corresponding to symmetric powers of the standard representation of quantum slN
Providing manageable definitions of these link homologies is one of the keys of the program aiming at categorifying quantum invariants of 3-manifolds
Summary
In [RW17], we provided a combinatorial evaluation of the foams underlying the (exterior) colored Khovanov-Rozansky link homologies [CK08a, CK08b, MS09, Sus, MSV09, Wu14, Yon11]. The present paper grew up as an attempt to provide a similar formula for foams underlying link homologies categorifying the Reshetikhin-Turaev invariants of links corresponding to symmetric powers of the standard representation of quantum slN. Whereas the definition of the link homologies can be made only using the language of foams and symmetric polynomial, the proof of invariance at the moment requires a more algebraic treatment This algebraic treatment uses Soergel bimodules and makes explicit the comparison with the work of Cautis [Cau17].(1) Cautis constructs a differential dN on the Hochschild homology of Soergel bimodules compatible with the differential of the Rickard complex such that the total homology provides a categorification of Reshetikhin-Turaev invariants of links corresponding to symmetric powers of the standard representation of quantum slN. We thank the referees for their numerous remarks which helped us to improve the paper exposition
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
