Grid presentation for Heegaard Floer homology of a double branched cover
Ghosh Hajra, Sayonita
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Heegaard Floer homology, introduced by Peter Ozsv ath and Zoltan Szab o, is an invariant of a closed oriented 3-manifold. Because of the complex nature of moduli spaces used in the defi nition, direct computations of this homology are di fficult to achieve. In 2006, Sarkar and Wang proposed an algorithm to compute the Heegaard Floer homology groups of a closed oriented 3-manifold in a purely combinatorial way. Ozsv ath, Stipsicz and Szab o improved on Sarkar-Wang's approach to show that using multi-pointed nice diagrams one can defi ne a combinatorial stabilized hat-version of Heegaard-Floer homology, and combinatorially prove its invariance. Stipsicz gave a description of the stabilized hat-version of Heegaard Floer homology for a double branched cover of 3-sphere branched over a link based on the grid presentation of the link. Here we present an algorithm to compute this stabilized version. We defi ne the extended grid homology groups associated to a link L and show combinatorially that the groups calculated by this algorithm are independent of certain choices made in constructing the extended grid diagram.