A proof of Dunfield-Gukov-Rasmussen Conjecture - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2022

A proof of Dunfield-Gukov-Rasmussen Conjecture

Anna Beliakova
  • Fonction : Auteur
Krzysztof K. Putyra
  • Fonction : Auteur
Louis-Hadrien Robert

Résumé

In 2005 Dunfield, Gukov and Rasmussen conjectured an existence of the spectral sequence from the reduced triply graded Khovanov-Rozansky homology of a knot to its knot Floer homology defined by Ozsv\'ath and Szab\'o. The main result of this paper is a proof of this conjecture. For this purpose, we construct a bigraded spectral sequence from the $\mathfrak{gl}_0$ homology constructed by the last two authors to the knot Floer homology. Using the fact that the $\mathfrak{gl}_0$ homology comes equipped with a spectral sequence from the reduced triply graded homology, we obtain our main result. The first spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules and a $\mathbb Z$-valued cube of resolutions model for knot Floer homology originally constructed by Ozsv\'ath and Szab\'o over the field of two elements. As an application, we deduce that the $\mathfrak{gl}_0$ homology as well as the reduced triply graded Khovanov-Rozansky one detect the unknot, the two trefoils, the figure eight knot and the cinquefoil.

Dates et versions

hal-03882784 , version 1 (02-12-2022)

Identifiants

Citer

Anna Beliakova, Krzysztof K. Putyra, Louis-Hadrien Robert, Emmanuel Wagner. A proof of Dunfield-Gukov-Rasmussen Conjecture. 2022. ⟨hal-03882784⟩
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