twist to the monodromy produces a tight contact structure.

Indeed, we need only choose $g$ large enough that $HF^{red}_{2g-1-h(\xi)}(-Y,{\mathfrak{s}}_\xi) = 0$. Analogously, we have

Fix an oriented 3-manifold $Y$. Then there is a rational number $r_Y\in {\mathbb{Q}}$ with the following property. Let $\xi$ be any contact structure on $Y$ with ${\mathfrak{s}}_\xi$ torsion and Hopf invariant $h(\xi)\leq r_Y$. Then $\xi$ becomes tight after adding a single right boundary-parallel twist to the monodromy of any supporting open book with connected binding.

In this corollary we simply choose $r_Y$ so that $HF^{red}_{d}(-Y, {\mathfrak{s}}) = 0$ for all torsion [[ ]{}]{}structures ${\mathfrak{s}}$, whenever $d \geq -r_Y$.

In a different direction, our results readily imply a connection between the “twist number” of a closed braid in $S^3$ and the reduced Khovanov homology of the link obtained as its closure. We thank John Baldwin and Liam Watson for bringing this to our attention.

\[braidthm\] Let $L$ be a link in $S^3$, and let $\hat{\beta}$ be any closed braid isotopic to $L$ and having an odd number of strands. Then $$|\tau(\hat{\beta})| \leq \dim_{{\mathbb F}}\widetilde{Kh}(-L) - |\det(L)| + 2.$$ Here $\tau(\hat{\beta})$ is the twist coefficient of $\beta$, viewed as an element in the mapping class group of the 