On the Naturality of the Spectral Sequence from Khovanov Homology to Heegaard Floer Homology

In [18], Ozsváth–Szabó established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link $\mathbb L\subset S^3$ and the Heegaard Floer homology of its double-branched cover. This relationship, extended in [19] and [4], was recast, in [5], as a specific instance of a broader connection between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhász in [7]. In the present work, we prove the naturality of the spectral sequence under certain elementary operations, using a generalization of Juhász’s surface decomposition theorem valid for decomposing surfaces geometrically disjoint from an imbedded framed link.