On the Heegaard Floer homology of branched double-covers

Let $L\subset S^3$ be a link. We study the Heegaard Floer homology of the branched double-cover $\mathrm{\Sigma }(L)$ of $S^3$, branched along $L$. When $L$ is an alternating link, $\text{HFa}$ of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose $E^2$ term is a suitable variant of Khovanov’s homology for the link $L$, converging to the Heegaard Floer homology of $\mathrm{\Sigma }(L)$.