Positroids, knots, and q,t-Catalan numbers
Abstract: Open positroid varieties are certain subvarieties of the Grassmannian that arise in the study of total positivity and have surprising applications in many areas of mathematics and physics. After reviewing some history and background, I will discuss our recent joint work with Thomas Lam relating the cohomology of these varieties and their point counts over finite fields to knot invariants such as the HOMFLYPT polynomial and Khovanov–Rozansky homology. In particular, we show that the bigraded Poincaré polynomials of top-dimensional open positroid varieties are given by rational q,t-Catalan numbers. No background on the above objects will be assumed.