one. If, for the map $$F: {\underline}{HF}^+(-Y_0(K), {{\mathbb F}}[C_n])\to HF^+(-Y_{-1/n}(K); {{\mathbb F}})$$ appearing in the surgery triangle, the restriction to the summand corresponding to ${\mathfrak{s}}_{1-g}$ is nontrivial, then the contact structure $\xi_n$ on $Y_{-1/n}$ is tight. The same is true if the genus of the fiber is one, provided we consider the surgery triangle with coefficients in the Novikov module $\Lambda_\omega$ associated to a $2$-form evaluating non-trivially on the fiber.

In Section \[trianglesec\] the map $F$ appearing in the surgery triangle is defined on the chain level by $$F(U^{-j}\zeta^k{\mbox{\bf x}}):= \sum_{\stackrel{\scriptstyle\psi\in \pi_2({\mbox{\bf x}},\Theta,{\mbox{\bf y}})}{\stackrel{{\mu(\psi)=0}}{{{n_p(\partial\psi)=-k \ \text{mod} \ n}}}}}\#{\mathcal{M}}(\psi)\cdot t^{\omega(\psi)}\cdot U^{n_w(\psi)-j}\cdot{\mbox{\bf y}}$$ (c.f. equation below). Here $t$ is the variable appearing in the Novikov ring $\Lambda_\omega$, which we set equal to 1 if the genus of the fiber is at least two. Comparing this to the definition of the cobordism-induced homomorphism $F^{{{\mathbb F}}[C_n]}_W$ in above, it is easy to check that there is a commutative diagram $$\begin{diagram}
\underline{CF}^+(-Y_0; {{\mathbb F}}[C_n]) & \rTo^{{F}_W^{{{\mathbb F}}[C_n]}} & \underline{CF}^+(-Y_{-1/n}; {{\mathbb F}}[C_n]) \\
\uTo^{\mathcal N} & \rdTo^{F} & \dTo^{\Pi}\\
CF^+(-Y_0; {{\mathbb F}}) &\rTo^{F_W}& CF^+(-Y_{-1/n}; {{\mathbb F}})
\end{diagram}$$ where $\mathcal N$ and $\Pi$ are chain maps induced by the coefficient ${{\mathbb F}}$-homomorphisms ${{\mathcal N}}: {{\mathbb F}}\to {{\mathbb 