A proof of Dunfield-Gukov-Rasmussen Conjecture

Abstract

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 Ozsváth and Szabó. The main result of this paper is a proof of this conjecture. For this purpose, we construct a bigraded spectral sequence from the $gl_{0}$ homology constructed by the last two authors to the knot Floer homology. Using the fact that the $gl_{0}$ 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 $Z$-valued cube of resolutions model for knot Floer homology originally constructed by Ozsváth and Szabó over the field of two elements. As an application, we deduce that the $gl_{0}$ homology as well as the reduced triply graded Khovanov-Rozansky one detect the unknot, the two trefoils, the figure eight knot and the cinquefoil.