TITLE: Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?
QUESTION [6 upvotes]: Consider the following question:
If $K\subset S^{3}$ is a nontrivial knot. Let $Y$ be the manifold obtained by doing $1/n$-surgery  ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ vanishes? 
The answer seems to be no. Because by  Gordon-Luecke's theorem, $Y$ is not $S^{3}$. While  as far as I know, people don't have any example of an integer homology 3-sphere other than $S^{3}$ having vanishing instanton Floer homology.
My question is: is there actually a rigorous proof of this problem? (for example, using instanton suture Floer homology?)

REPLY [3 votes]: Kronheimer and Mrowka first proved that 1-surgery on a knot has non-trivial instanton Floer homology using Witten's conjecture (see their paper http://arxiv.org/abs/math/0311489).  They later proved this using sutured instanton Floer homology as you suggested (see Section 7 of http://arxiv.org/abs/0807.4891).
As far as I know, for $n$ at least 2, this is unknown in general.  K-M's first proof of Property P uses a long exact sequence in instanton Floer homology relating the instanton Floer homologies of $S^3$, $S^3_1(K)$, and $S^3_0(K)$.  Some total speculation would be to try to construct a long exact sequence relating the instanton Floer homologies of $S^3$, $S^3_{1/n}(K)$, and $S^3_0(K)$ with some sort of twisted coefficients (analogous to Theorem 9.14 in http://arxiv.org/abs/math/0105202) and try to repeat their argument.
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