I will start by giving an introduction to polynomial knot invariants, as well as to Khovanov's more recent idea of knot homology. The goal is to find bi- and tri-graded vector spaces whose graded Euler characteristics are classical polynomial knot invariants (like the Jones or HOMFLYPT polynomial). I will then explain how HOMFLYPT homology can be given a transparent construction using some heavy machinery from geometric representation theory. This gives a bridge between link homology and techniques which have been developed for studying the characters of finite groups of Lie type.