Homology 3-sphere with a unique Stein-fillable contact structure

Question

Are there any known examples of oriented integer homology 3-spheres $Y$ (besides $S^3$) which have exactly one Stein-fillable contact structure up to isotopy? Failing that, what are the known examples with the smallest number of Stein-fillable contact structures where one exists? Note here I am only counting Stein-fillable contact structures not other contact structures on $Y$.

There are situations where I'd like to be able to compute bounds on the maximum of the adjunction number $ad(K)=tb(K)-1+|r(K)|$ for $K$ a knot in $Y$ allowing both the Legendrian representative of $K$ to vary as well as the contact structure. I was wondering if there were cases where this was at all tractable. I have not spent much thought on the classification of contact structures, so I do not know how hard this should be.

Answer 1

One example you might want to start with is the Poincaré homology sphere. It is a Seifert fibred space over $S^2$ with three singular fibres $M(\frac{1}{2}, - \frac{1}{3}, -\frac{1}{5})$ supporting exactly one Stein fillable contact structure. Note that the the Poincaré homology sphere with reversed orientation $\overline{\Sigma}(2,3,5)$ was the first example of a three-manifold with no positive tight contact structure (Etynre and Honda, 1999).

I'm not sure if a proof ever appeared in the literature, but the fact that the Poincaré homology sphere supports one Stein fillable contact structure is not hard to prove using the techniques from Etnyre and Honda. A proof follows exactly the lines further examples were classified, which gives a second example:

The Brieskorn homology sphere $\overline{\Sigma}(2,3,11)$ supports exactly one Stein fillable contact structure. Again, this is a Seifert fibred space over $S^2$ with three singular fibres $M(-\frac{1}{2}, \frac{1}{3}, \frac{2}{11})$. The proof of this was written up by Paolo Ghiggini in 2003.

As Marco Golla remarks, you might be able to come up with many more examples from the classification results on Seifert fibred spaces by Ghiggini, Lisca, Stipsicz, Wu. I hope the two concrete examples are of any help for you.