it is a well-known theorem in combinatorics that everything is counted by the catalan numbers. these numbers are ubiquitous in mathematics, often appearing where you least expect them — such as in the homflypt polynomial of torus knots. in this talk, we’ll take a journey into algebraic combinatorics by pursuing the question: why do the torus knots know about catalan numbers? along the way, we’ll encounter braid groups, hecke algebras, complex reflection groups, and more! this talk is based on a paper by pavel galashin, thomas lam, minh-tam quang trinh, and nathan williams.