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Szabo in A geometric spectral sequence in Khovanov homology gives a combinatorial candidate for what an explicit calculation of the spectral sequence of branched double covers should yield. In other words he gives a conjectural combinatorial model for HF-hat of branched double covers of links.

However the input for his algorithm is not a bare link diagram; it's a "decorated" diagram. The decoration is a choice of orientations for the arcs which connect the two strands of the link at crossings. Szabo says in the paper that these decorations are analogues of the extra structure given by Heegaard diagrams and almost complex structures in Heegard–Floer homology. But I don't understand how.

So, my question is, what is the analogue of Szabo's decorations in Heegaard–Floer homology? (Since the two theories are expected to be isomorphic, there should be such an analogue.) I suspect this has to do with Lipshitz's cylindrical reformulation but am not sure. By the way, this whole story is over Z/2 so signs are not involved.

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I'm pretty sure those orientations on arcs correspond to an orientation of the surgery link in the construction of Ozsvath-Szabo's original spectral sequence from Khovanov homology to HF of the branched double cover.

More precisely, to construct this spectral sequence, you start with a link projection, and take the branched double cover. Each crossing is contained in a small ball with two strands, whose branched double cover is the solid torus. The arc connecting these two strands lifts to a knot in the branched double cover (the core of the solid torus), and taking all the crossings together you get a link $L$.

Thus, choosing orientations at each crossing, as in Szabo's paper, is the same as orienting this link $L$. In the end, the spectral sequence is independent of these orientations, but you have to choose them at the beginning.

These orientations are also used in constructing odd Khovanov homology (http://arxiv.org/abs/0710.4300), for the same reason (it's the sign convention you naturally get on the Khovanov complex by considering it as the $E_1$ term of the spectral sequence to HF of the branched double cover).

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  • $\begingroup$ Thank you. However the decoration in odd Khovanov homology seems to be to determine the signs and the authors say if one considers $\Z/2$ coefficients one does not need those decorations. Also the link surgeries spectral sequence (as described in Ozsvath and Szabo's branched double cover paper), doesn't seem to depend on the orientation of the link. (Although in Heegaard-Floer theory one assumes the manifold to be oriented and this together with the framing gives an orientation of the link.) $\endgroup$
    – Reza Rezazadegan
    Commented May 23, 2013 at 16:16
  • $\begingroup$ Right, odd Khovanov homology is just another way of lifting $\mathbb{Z}/2$ Khovanov homology to the integers, but I think from this HF-of-branched-double-cover perspective it's more of a coincidence that when you reduce mod 2 you don't need the crossing orientations to define the Khovanov edge maps. The orientation you mention is the one I was thinking of, and I think it's still necessary to define the maps in the spectral sequence on the chain level (even though the resulting object is independent of them). $\endgroup$
    – Andy Manion
    Commented May 23, 2013 at 18:46

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