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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...

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  • $\begingroup$ It definitely holds whenever $K$ is (Khovanov-reduced, I believe) homologically thin, and when $K$ is a 2-bridge knot. The first case follows from the spectral sequence defined by Ozsvath-Szabo, the second from the classical fact that double covers branched over 2-bridge knots are lens spaces. $\endgroup$
    – Marco Golla
    Commented May 22, 2014 at 6:39

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The paper of Manolescu and Owens "A concordance invariant from the floer homology of double branched covers" seems to answer your question. They compute many examples of the Froyshov/d-invariant in their paper, using software of Saso Strle. It's available on the arXiv. In that case it's at least reasonably-computable.

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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).

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Let assume $Y$ is a closed oriented rational homology sphere.

The set of spin$^c$ structure of $Y$ is in one to one correspondence with $H_1(Y,\mathbb{Z})$. So if $H_1(Y,\mathbb{Z}) \neq 0$ then there are more than one spin$^c$ structure.

The question then becomes: When does $\widehat{HF}(Y,\mathfrak{s}) \cong \mathbb{Z}$ for each spin$^c$ structure $\mathfrak{s}$?

Three-manifolds with such property are called $L$-spaces. Double branched cover of quasi-alternating knots (in $S^3$) are L-spaces. I think your question is a work in progress. For some other examples see Steven Sivek comment below.

Note also that the collection of quasi-alternating knots contains the collection of alternating knots, and 2-bridge knots are examples of alternating knots.

In the special case $H_1(Y,\mathbb{Z}) = 0$, in which situation there is a unique spin$^c$ structure, it is conjectured that $S^3$ and the Poincar\'e homology sphere ($\Sigma=S^3/I$ where $I$ is the full icosahedral group) are the only such 3-manifold. We know that $d(S^3)=0$ and that $d(\Sigma)=\pm 2$ depending on the orientation you chose. For the general case I think knowing exactly the $d$-invariant is still an open problem except when $Y$ is a lens space in which case there is a formula for each $d$ invariant.

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  • $\begingroup$ There are actually lots of non-QA knots whose branched double covers are L-spaces. See section 6.1 of arxiv.org/pdf/1205.5261.pdf for some discussion and explicit examples, including the $P(p_1,\dots,p_n,-q) $ pretzel knots where $p_i,q>0$ and $q = \min(p_1,\dots,p_n)$. $\endgroup$
    – Steven Sivek
    Commented Jun 13, 2014 at 3:08

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