On the invariance of the Dowlin spectral sequence

TL;DR

The paper proves that the higher pages $E_k$ of Dowlin's spectral sequence, which begins at $E_2\cong\overline{Kh}(L)$ and converges to $\widehat{HFK}(m(L))$, are link invariants for all $k\ge 3$. By establishing invariance under vertex relabeling and MOY moves, and then verifying Reidemeister II/III, stabilization, and conjugation moves via Gaussian-elimination techniques and MOY decompositions, the authors show that the entire spectral sequence structure provides a robust family of link invariants $\{E_k(L)\}_{k\ge 2}$. The construction leverages matrix-factorizations, regular-sequence conditions, and a carefully controlled cube of resolutions to transport Khovanov data to knot Floer data across diagram changes. The results deepen the interplay between Khovanov and knot Floer homologies and open avenues for distinguishing knots with identical $\overline{Kh}$ and $\widehat{HFK}$ via higher-page information, with potential applications to transverse invariants and $s$/$\tau$-type correspondences.

Abstract

Given a link $L$, Dowlin constructed a filtered complex inducing a spectral sequence with $E_2$-page isomorphic to the Khovanov homology $\overline{Kh}(L)$ and $E_\infty$-page isomorphic to the knot Floer homology $\widehat{HFK}(m(L))$ of the mirror of the link. In this paper, we prove that the $E_k$-page of this spectral sequence is also a link invariant, for $k\ge 3$.

Figures (30)

• Figure 2.1: The different types of vertices in a partially singular braid diagram.
• Figure 2.2: Other features that can occur in a braid diagram.
• Figure 2.3: The $0$- and $1$-resolutions of positive and negative crossings.
• Figure 2.4: The local edge labels around a vertex.
• Figure 2.5: Unoriented smoothing of a crossing.

Theorems & Definitions (49)

• Theorem 1.1
• Definition 2.1
• Definition 2.2: nate
• Remark 2.3
• Definition 2.4
• Theorem 2.5: nate
• Proposition 2.6
• Definition 2.7
• proof : Proof of \ref{['prop:psbd-existence']}
• Theorem 2.8