According to Theorem 1.7 of Mark-Tosun's [paper](https://arxiv.org/pdf/1603.07710.pdf), 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.



Ozsváth-Szabó's contact invariant on the Brieskorn sphere $\Sigma(2,3,6m+1)$
(Note: I might have screwed up orientations, so take everything with a grain of salt.) I will write an argument for the case $m=1$, showing that the reduced contact invariant $c^{\rm red}(\xi)$ is non-zero. I think that the argument(s) can be tweaked to work for $m>1$, as well, but I won't try to write it down properly.


Let me suppress some things from the notation: I'll call $\Sigma = \Sigma(2,3,7)$, and let $\xi$ denote either of $\xi\_0$ or $\xi\_1$. (The statement is invariant under self-diffeomorphisms of $\Sigma$.)


Mark and Tosun give an explicit surgery presentation of $\xi$ as Legendrian surgery along a right-handed trefoil with Thurston-Bennequin $0$. This gives a Stein cobordism $W: S^3 \leadsto \Sigma$, and the cobordism map $F: = F^+\_{\overline W}: HF^+(-\Sigma) \to HF^+(-S^3)$ maps $c(\xi)$ to $c(\xi\_{\rm std})$.
The key point is showing that $F$ vanishes on the tower in $HF^+(\Sigma)$ (i.e. it vanishes for sufficiently large degrees). This can be seen in two ways.


The first is to show that spin$^c$ structures on $W$ come in conjugate pairs, and all their contributions cancel in pairs on the tower. (Note that they can -and do, a posteriori- move elements outside the tower.)


The second is that $F$ fits into an exact triangle:
$$ \dots \to HF^+(-\Sigma\_0) \to HF^+(-\Sigma) \to HF^+(-S^3)\to \dots$$
where $\Sigma\_0$ is the 0-surgery along the (right-handed) trefoil.
It's easy to see (basically by rank count) that the map $HF^+(-\Sigma\_0) \to HF^+(-\Sigma)$ must be surjective onto the tower, which must then be killed by $F$.


Either way, if $c^{\rm red}(\xi)=0$, then $F(c^{\rm red}(\xi)) = 0 \neq c(\xi\_{\rm std}))$.


