  We relate the mixed Hodge structure on the cohomology of open positroid
varieties (in particular, their Betti numbers over $\mathbb{C}$ and point
counts over $\mathbb{F}_q$) to Khovanov--Rozansky homology of associated links.
We deduce that the mixed Hodge polynomials of top-dimensional open positroid
varieties are given by rational $q,t$-Catalan numbers. Via the curious
Lefschetz property of cluster varieties, this implies the $q,t$-symmetry and
unimodality properties of rational $q,t$-Catalan numbers. We show that the
$q,t$-symmetry phenomenon is a manifestation of Koszul duality for category
$\mathcal{O}$, and discuss relations with open Richardson varieties and
extension groups of Verma modules.
