Khovanov homology, sutured floer homology and annular links

In [28], Lawrence Roberts, extending the work of Ozsva ́th and Szabo ́ in [22], showed how to associate to a link L in the complement of a fixed unknot B ⊂ S³ a spectral sequence whose E² term is the Khovanov homology of a link in a thickened annulus defined by Asaeda, Przytycki and Sikora in [1], and whose E^∞ term is the knot Floer homology of the preimage of B inside the double-branched cover of L. In [6], we extended [22] in a different direction, constructing for each knot K ⊂ S³ and each n n Ie{cyrillic, ukrainian} 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 Juha ́sz's sutured Floer homology [8] and show that the spectral sequence of [6] is a direct summand of the spectral sequence of [28].