In , P. Ozsváth and Z. Szabóconstructed a spectral sequence computing the Heegaard-Floer homology HF(YL) where YL is the result of surgery on a framed link, L, in Y. The terms in the E 1-page of this spectral sequence are Heegaard-Floer homologies of surgeries on L for other framings derived from the original. They used this result to analyze the branched double cover of a link L ⊂ S 3 where it was possible to give a simple description of all the groups arising in the E 1-page. The result is a spectral sequence, over F2, with E 2 page given by the reduced Khovanov homology of L and converging in finitely many steps to HF(−Σ(L)), where Σ(L) is the branched double cover of S 3 over L. Several years later, in ,  adjusted this argument to a setting where the spectral sequence started at refinement of Khovanov homology, and converged to a knot Floer homology. This facilitated the analysis of the knot Floer homology of certain fibered knots. Recently, Olga Plamanevskaya first in  used this approach to show that the contact invariant of certain open books was non-vanishing. By generalizing from the double branched cover picture, she and John Baldwin, , were able to extend the argument to more general open books. In addition, Eli Grigsby and Stefan Wehrli,  found a different direction in which to generaliz
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