From Combinatorics to Knot Theory (and Back Again)

To be held at MATX 1100 and on Zoom: https://ubc.zoom.us/j/68285564037?pwd=R2ZpLy9uc2pUYldHT3laK3orakg0dz09
Meeting ID: 682 8556 4037
Passcode: 636252

Reception and refreshments at 15:30 in the math department lounge, MATH 126.

Catalan numbers are among the most ubiquitous objects in mathematics, arising naturally in combinatorics, representation theory, geometry, and many other areas. Although there are various polynomial generalizations of these numbers, particularly fruitful are the so-called $(q,t)$-Catalan polynomials. Among many other things, these polynomials provide a direct link between combinatorial objects, such as Dyck paths and parking functions, and the Khovanov-Rozansky homology (a particular homological link invariant) of so-called torus knots. In this talk, I will explain some of the fascinating connections—both known and conjectural—between various Catalan objects and knot theory. I will also present new families of Catalan polynomials, constructed by my collaborators and me, that generalize previous formulations and provide new insights into the Khovanov-Rozansky homology for the larger family of Coxeter knots. Along the way, we provide the first proof of a conjecture by Oblomkov-Rasmussen-Shende for a family of cabled torus knots and establish a double affine Hecke algebra analog of the celebrated Shuffle Theorem.