A proof of Dunfield–Gukov–Rasmussen conjecture

  • Anna Beliakova

    Universität Zürich, Zürich, Switzerland
  • Krzysztof Putyra

    Universität Zürich, Zürich, Switzerland
  • Louis-Hadrien Robert

    Université Clermont Auvergne, Aubière, France
  • Emmanuel Wagner

    Université Paris Cité, Paris, France
A proof of Dunfield–Gukov–Rasmussen conjecture cover
Download PDF

A subscription is required to access this article.

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