  We give a numerical condition for right-handedness of a dynamically convex
Reeb flow on $S^3$. Our condition is stated in terms of an asymptotic ratio
between the amount of rotation of the linearised flow and the linking number of
trajectories with a periodic orbit that spans a disk-like global surface of
section. As an application, we find an explicit constant $\delta_* < 0.7225$
such that if a Riemannian metric on the $2$-sphere is $\delta$-pinched with
$\delta > \delta_*$, then its geodesic flow lifts to a right-handed flow on
$S^3$. In particular, all finite non-empty collections of periodic orbits of
such a geodesic flow bind open books whose pages are global surfaces of
section.
