Given a braid, one can associate to it a collection of “categorified” braid invariants in two apparently different ways: “algebraically,” via the representation theory of Uq(sl2) (using ideas of Khovanov and Seidel) and “geometrically," via Floer theory (specifically, Ozsvath-Szabo's Heegaard Floer homology package as extended by Lipshitz-Ozsvath-Thurston). Both collections of invariants are strong enough to detect the trivial braid. I will discuss what we know so far about the connection between these invariants, focusing on the relationship between the representation theory and the Floer theory. This is joint ongoing work with Denis Auroux and Stephan Wehrli.