TITLE: Heegaard Floer Homology of double branched cover
QUESTION [5 upvotes]: The question is the following:
Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: 
When do we have $\widehat{HF}(Y,\mathfrak{s})\cong \mathbb{Z}$? If yes, what do we know about the d-invariant?
I know that this is true for quasi-alternating knot, however, I don't know any other examples...

REPLY [3 votes]: John Baldwin said something about both questions in this paper. He considers closures of 3-braids and classifies which of these have an $L$-space branched double cover (Theorem 4.1). He then computes the correction term for the spin structure (Theorem 5.1); he also seems to say that there are no new results in this last computation (see the discussion before the statement of the theorem).