Szabo in http://arxiv.org/abs/1010.4252 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 "dcorated" diagram. The decoration is a choice of orientations for the arcs which connect the two strands of the link at crossings. Szabo sais 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.