Khovanov homology, sutured Floer homology, and annular links

TL;DR

This paper explores the relationship between Khovanov homology, sutured Floer homology, and annular links, extending and connecting previous spectral sequence constructions to deepen understanding of link invariants.

Contribution

It reinterprets Roberts' spectral sequence within sutured Floer homology and shows it as a summand of a previously constructed spectral sequence, unifying different approaches.

Findings

Roberts' spectral sequence is a summand of the authors' spectral sequence.

The work bridges Khovanov homology and sutured Floer homology frameworks.

Provides a new perspective on link invariants in 3-manifolds.

Abstract

Lawrence Roberts, extending the work of Ozsvath-Szabo, showed how to associate to a link, L, in the complement of a fixed unknot, B, in S^3, a spectral sequence from the Khovanov homology of a link in a thickened annulus to the knot Floer homology of the preimage of B inside the double-branched cover of L. In a previous paper, we extended Ozsvath-Szabo's spectral sequence in a different direction, constructing for each knot K in S^3 and each positive integer n, a spectral sequence from Khovanov's categorification of the reduced, n-colored Jones polynomial to the sutured Floer homology of a reduced n-cable of K. In the present work, we reinterpret Roberts' result in the language of Juhasz's sutured Floer homology and show that our spectral sequence is a direct summand of Roberts'.