When we think of quantum invariants, we usually think of the Jones polynomial or of the coloured HOMFLYPT. But (arguably) the simplest example of a quantum invariant of a knot or link is its Alexander polynomial. From the beginning, the central problem in the study of quantum invariants has been what do they mean topologically? The Alexander polynomial has clear algebraic topological meaning as the order of the Alexander module (first homology of the infinite cyclic cover as a module over the group of deck transformations). Can people conceptually explain (in terms of both physics and mathematics) why the representation theory of certain small quantum groups naturally gives rise to this quantity? Computationally I can understand it, but not conceptually.
A somewhat related question was already asked here.