bound on the twist coefficient of a fibered knot, given the homotopy data of its associated contact structure.

\[thm3\] Let $\xi$ be a contact structure on $Y$ whose associated [[ ]{}]{}structure ${\mathfrak{s}}_\xi$ is torsion, and let $(S, \phi)$ be any genus $g$ open book supporting $\xi$ with connected binding. Then the twist coefficient of $\phi$ satisfies $$-1 - \dim_{{\mathbb F}}HF^{red}_{2g-1-h(\xi)}(-Y, {\mathfrak{s}}_\xi) \,\leq\, \tau(\phi)\, \leq\, 1 + \dim_{{\mathbb F}}HF^{red}_{-h(\xi)}(-Y,{\mathfrak{s}}_\xi).$$

A slightly sharper version is given in Corollary \[twistboundcor\] below, which also applies when ${\mathfrak{s}}_\xi$ is non-torsion. As before, the bounds on twist number come from estimating the number of boundary twists which, when added to the monodromy, is sufficient to obtain a tight contact structure. These estimates, combined with the dependence on the page genus in Theorem \[thm3\] and the remark in the previous paragraph, yield a surprising corollary. It indicates that “most" open books which support a given contact structure yield a tight structure after adding a single right-handed Dehn twist along the boundary.

Let $\xi$ be a contact structure with torsion Euler class, and let $g$ be sufficiently large. Then for [*any*]{} genus $g$ open book decomposition which supports $\xi$ (with connected binding), adding a single right-handed, boundary-parallel Dehn 