John Baez Lots of cool papers have been appearing which I've been neglecting in my attempts to write expository stuff about string theory, lattices, category theory, and all that. Ever since Jones discovered the Jones polynomial invariant of knots, an amazing story has been unfolding about the relation between algebra and 3-dimensional topology.
All quantum groups give rise to invariants of knots, links, and tangles.
This is a kind of quantum field theory you can define on a 3-dimensional spacetime that you've triangulated, i.e., chopped up into tetrahedra. Ponzano and Regge had tried to set up 3-dimensional quantum gravity this way previously, but there were problems getting the sum to converge. The neat thing about the quantum group is that you only sum over spins less than some fixed spin depending on the value of hbar. Since the sums are finite, they automatically converge. It turns out that in these Turaev-Viro theories you are not actually taking advantage of all the structure of the quantum group.