A proof of Dunfield-Gukov-Rasmussen Conjecture

TL;DR

The paper proves the Dunfield–Gukov–Rasmussen conjecture by constructing a bigraded spectral sequence from the algebraic side $H^{\mathfrak{gl}_0}$ to knot Floer homology $\widehat{HFK}$, threading through a prior $HHH^{red}\to H^{\mathfrak{gl}_0}$ sequence and a new $q$-deformed cube of resolutions. Central to the approach is the $qAG$ framework, a $\mathbb{Z}$-valued, coefficient-smart complex that interpolates between the algebraic and geometric knot invariants via a $(q\mapsto 1)$ Bockstein, whose $E_1$-page is $H^{\mathfrak{gl}_0}$ and whose limit is $\widehat{HFK}$. Consequently, the authors obtain a two-stage spectral sequence linking $HHH^{red}$, $H^{\mathfrak{gl}_0}$, and $\widehat{HFK}$, establishing a rank inequality that yields a DGR-type conjecture in characteristic zero and enabling knot-detection results for the involved homologies. The work also demonstrates that $H^{\mathfrak{gl}_0}$ and $HHH^{red}$ detect key knots (unknot, trefoils, figure-eight, cinquefoil), with these detections translatable to knot Floer homology through the constructed spectral sequences. Overall, the paper forges a concrete bridge between algebraic and geometric knot invariants, providing new computational tools and deep structural insight into how these theories bound and reflect knot topology.

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 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 $\mathfrak{gl}_0$ 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 Ozsváth and Szabó 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.

Figures (26)

• Figure 1: Pictorial definition of $\lambda^c$, $\lambda^t$ and $\widehat{\lambda}$.
• Figure 2: A split and a merge vertex in a web.
• Figure 3: A directed planar web of index 4 (on the left) and its lamination (on the right).
• Figure 4: The three local models for a foam.
• Figure 5: Local models for all singularities of directed foams, together with their degrees.

Theorems & Definitions (142)

• Proposition A
• Conjecture 1
• Theorem B
• Theorem : RW3
• Theorem C: DGR Conjecture
• Corollary D
• proof
• Corollary E
• proof
• Proposition 2.2