Positroids, knots, and q,t-Catalan numbers

We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over C and point counts over Fq) 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 O, and discuss relations with open Richardson varieties and extension groups of Verma modules.