A proof of Dunfield–Gukov–Rasmussen conjecture
Anna Beliakova
Universität Zürich, Zürich, SwitzerlandKrzysztof Putyra
Universität Zürich, Zürich, SwitzerlandLouis-Hadrien Robert
Université Clermont Auvergne, Aubière, FranceEmmanuel Wagner
Université Paris Cité, Paris, France

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 homology constructed by the last two authors. The homology comes equipped with a spectral sequence from the reduced triply graded homology, and here we construct the other spectral sequence, from the 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 -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 homology and the reduced triply graded Khovanov–Rozansky homology detect the unknot, the two trefoils, the figure eight knot and the cinquefoil.
Cite this article
Anna Beliakova, Krzysztof Putyra, Louis-Hadrien Robert, Emmanuel Wagner, A proof of Dunfield–Gukov–Rasmussen conjecture. J. Eur. Math. Soc. (2025), published online first
DOI 10.4171/JEMS/1626