Liam Watson for interesting conversations and valuable input. Matt Hedden was partially supported by NSF grant DMS-0906258, NSF CAREER grant DMS-1150872, and an Alfred P. Sloan Research Fellowship. Thomas Mark was partially supported by NSF grants DMS-0905380 and DMS-1309212.

Proof of Theorems \[thm1\] and \[thm2\] {#proofsec}
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We work in characteristic two throughout, and let ${{\mathbb F}}={{\mathbb Z}}/2{{\mathbb Z}}$. This is for simplicity, and all our arguments could be made with ${{\mathbb Z}}$ in place of ${{\mathbb F}}$.

0.1in In this section we prove Theorem \[thm2\], from which Theorem \[thm1\] will follow easily. More precisely, we show that adding $$N_Y = {\textstyle 1\over 2}(\mathrm{dim}_{{\mathbb F}}\widehat{HF}(Y) - |\mathrm{Tor}_{\mathbb{Z}}\ H_1(Y;{\mathbb{Z}})|)$$ right-handed Dehn twists to the boundary of any open book decomposition $(S,\phi)$ of $Y$ will produce an open book decomposition for a tight contact structure. The key observation is that the manifold specified by $(S,\phi\circ t_\partial^n)$ is homeomorphic to $Y_{-1/n}(K)$, where $K=\partial S$ is the binding of the open book, viewed as a knot in $Y$. Let $\xi_{n}$ denote the contact structure on $Y_{-1/n}(K)$ induced by $(S,\phi\circ t_\partial^n)$. Our strategy is as follows

1.  Observe that to show $\xi_{n}$ is tight, it suffices by [@Contact Theorem 1.4] to show that its contact 