Khovanov homology, sutured Floer homology and annular links

Abstract

In [arXiv:0706.0741], Lawrence Roberts, extending the work of Ozsváth and Szabó in [Adv. Math 194 (2005) 1-33], showed how to associate to a link $\mathbb{L}$ in the complement of a fixed unknot$B\subset S^3$ a spectral sequence whose $E^2$ term is the Khovanov homology of a link in a thickened annulus defined by Asaeda, Przytycki and Sikora in [Algebr. Geom. Topol. 4 (2004) 1177-1210], and whose $E^{\infty }$ term is the knot Floer homology of the preimage of $B$ inside the double-branched cover of $\mathbb{L}$.

In [Adv. Math. 223 (2010) 2114-2165], we extended the aforementioned Ozsváth–Szabó paper in a different direction, constructing for each knot $K\subset S^3$ and each $n\in {\mathbb{Z}}_{+}$, 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 [Algebr. Geom. Topol. 6 (2006) 1429–1457] and show that the spectral sequence of [Adv. Math. 223 (2010) 2114-2165] is a direct summand of the spectral sequence of Roberts’ paper.