2022/10/03 by Anna Beliakova, Beliakova, Anna, Louis‐Hadrien Robert +4 · 2 citations
Mathematics · #18G40 #55U20 #57K18 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.48550/arxiv.2210.00878
openalex publication_date 2022/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathfrakgl0 homology constructed by the last two authors to the knot Floer homology. Using the fact that the \mathfrakgl0 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 Ozsváth and Szabó over the field of two elements. As an application, we deduce that the \mathfrakgl0 homology as well as the reduced triply graded Khovanov-Rozansky one detect the unknot, the two trefoils, the figure eight knot and the cinquefoil.