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Notes on the Heegaard-Floer link surgery spectral sequence

By Lawrence Roberts


In [8], 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 [12], [13] 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 [11] 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, [1], were able to extend the argument to more general open books. In addition, Eli Grigsby and Stefan Wehrli, [4] found a different direction in which to generaliz

Year: 2013
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