TITLE: Ozsváth-Szabó's contact invariant on the Brieskorn sphere $\Sigma(2,3,6m+1)$
QUESTION [5 upvotes]: According to Theorem 1.7 of Mark-Tosun's paper, the Brieskorn sphere $\Sigma(2,3,6m+1)$ admits two tight contact structure $\xi_{i}\ (i=0,1)$. They are both Stein fillable and they are contactomorphic (but not isotopic).
My question:

Consider the Ozsváth-Szabó contact invariant $c(\xi_{i})\in HF^{+}(-\Sigma(2,3,6m+1))$. We know it is non-zero by Stein fillability, but do we know whether its image under the natural map 
  $$
HF^{+}(-\Sigma(2,3,6m+1))\rightarrow HF^{\text{red}}(-\Sigma(2,3,6m+1))
$$
  is non-zero?

Such result can be shown, e.g., if $\xi_{i}$ bounds a non negative definite Stein domain.

REPLY [3 votes]: Brieskorn-Pham calculated the signature of the smoothing of the Milnor fiber of that Brieskorn singularity (in this case the smoothing of the complex singularity $x^2+y^3+z^{6m+1}=0$ or the symplectic branched double cover over the smoothing of the algebraic surface bounding $T(3,7)$ ). In the case of $M(2,3,6m+1)$, this is non-definite. See remark 4.6 of http://www.maths.ed.ac.uk/~aar/papers/nemethi1.pdf for Brieskorn's formula.
In the case of $\Sigma(2,3,6m+1)$, the contact structure has homotopy type $\theta=-2$. As $c_1=0$ for $M(2,3,6m+1)$ (this can be seen by the double cover representation), Gompf's formula for $\theta$, $c_1^2-3\sigma(X)-2\chi(X)=\theta$ gives 
$$\sigma(M(2,3,6m+1))=-8m$$