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
in the complement
of a fixed unknot a
spectral sequence whose
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
term is the knot Floer homology of the preimage of
inside the
double-branched cover of .
In [Adv. Math. 223 (2010) 2114-2165], we extended the aforementioned
Ozsváth–Szabó paper in a different direction, constructing for each knot
and
each ,
a spectral sequence from Khovanov’s categorification of the reduced,
–colored
Jones polynomial to the sutured Floer homology of a reduced
–cable
of . 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.
Keywords
Heegaard Floer homology, Khovanov homology, link
invariants, branched covers
