Year: 2024
Type: article
Abstract: Given a link L, Dowlin constructed a filtered complex inducing a spectral sequence with E 2 -page isomorphic to the Khovanov homology Kh.L/ and E 1 -page isomorphic to the knot Floer homology 1 HFK.m.L// of the mirror of the link.We prove that the E k -page of this spectral sequence is also a link invariant, for k 3. 57K18 1 IntroductionDowlin [2024] associated a filtered chain complex to a link L. The spectral sequence this filtered complex gives rise to has E 2 -page isomorphic to the (reduced) Khovanov homology Kh.L/ and converges to the knot Floer homology 1 HFK.m.L// of the mirror of the link.The fact that the E 2 -and E 1 -pages of the spectral sequence are link invariants, independent of the diagram used to construct the filtered complex, suggests that the same may be true of all the higher pages of the spectral sequence.This is the main result of this paper.Theorem 1.1 For k 2, the E k -page of Dowlin's spectral sequence does not depend on the diagram used to construct the filtered complex, and is thus a link invariant.This theorem provides a whole family of link invariants fE k .L/g 1 kD2 .The invariance of these higher pages of the Dowlin spectral sequence helps us further decipher the connection between Khovanov homology and knot Floer homology.This result opens several research directions.The first is to find knots (or families of knots) which have the same Khovanov homology and knot Floer homology, but are distinguished by these higher page invariants.The ranks of Khovanov homology and knot Floer homology tend to coincide for knots with few crossings [Rasmussen 2005], so finding such examples may be computationally difficult.A second direction is to consider implications in the study of transverse links.Plamenevskaya [2006] identified an invariant of transverse links .L/ 2 Kh.L/, which we can think of as residing in the E 2 page of the Dowlin spectral sequence.One could hope to define a countable family of transverse link invariants f k .L/g 1 kD2 by taking the image of on each higher page E k for k 2, in the style of Baldwin [2011].It might prove interesting to compare these invariants, especially the image of on the
Source: Algebraic & Geometric Topology
Authors: Samuel Tripp, Zachary Winkeler
Institutions: Worcester Polytechnic Institute, Smith College
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4405954340