Nicolle González: From combinatorics to knot theory and back again

Nicolle González (UC Berkeley)

The combinatorics of (q,t)-Catalan polynomials and related objects is known to have deep connections to geometry, representation theory, and knot theory. In particular, the (q,t)-Catalan polynomials arise as the hook components in the celebrated Shuffle theorem of Carlsson-Mellit which combinatorially computes the Frobenius character of the space of diagonal harmonics. At the same time, the (q,t)-Catalan polynomials arise as the lowest degree part of the Khovanov-Rozanksy homology of certain torus knots. Khovanov-Rozanksy homology is a distinguished knot-homology theory that is notoriously difficult to compute. In this talk I will describe how both the combinatorial and topological components in this story are related and introduce two new families of polynomials that generalize the $(q,t)$-Catalan polynomials in different directions. Along the way, these polynomials will not only compute the KR-homology for larger previously unknown families of knots and prove some open conjectures, they will also pose new unexplored questions by changing the representation theory that underlies them.