A proof of Dunfield-Gukov-Rasmussen Conjecture

TL;DR

This paper proves the Dunfield-Gukov-Rasmussen conjecture by constructing a spectral sequence linking Khovanov-Rozansky homology to knot Floer homology, revealing new connections between these knot invariants.

Contribution

It provides the first proof of the conjectured spectral sequence, using quantum traces and a cube of resolutions model, advancing understanding of knot homologies.

Findings

Spectral sequence from $rak{gl}_0$ homology to knot Floer homology constructed.

$rak{gl}_0$ and Khovanov-Rozansky homologies detect specific knots.

New tools include quantum traces of foams and singular Soergel bimodules.

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 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.