A proof of Dunfield–Gukov–Rasmussen conjecture

Abstract

In 2005, Dunfield, Gukov and Rasmussen conjectured an existence of a differential 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 a suitably updated version of their conjecture: we show that the reduced triply graded homology is related to knot Floer homology by two spectral sequences, going through the intermediate ${\mathfrak{gl}}_0$ homology constructed by the last two authors. The ${\mathfrak{gl}}_0$ homology comes equipped with a spectral sequence from the reduced triply graded homology, and here we construct the other spectral sequence, from the ${\mathfrak{gl}}_0$ homology to knot Floer homology. The new spectral sequence is of Bockstein type and arises 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 Ozsváth–Szabó over the field of two elements. As an application, we deduce that both the ${\mathfrak{gl}}_0$ homology and the reduced triply graded Khovanov–Rozansky homology detect the unknot, the two trefoils, the figure eight knot and the cinquefoil.