How an ordinary Hopf algebra yields a 3-manifold invariant

There are many constructions of quantum 3-manifold invariants, some of them complicated and some of them not so complicated. I will describe a relatively less popular invariant of 3-manifolds whose definition and proof of invariance is among the simplest in the field of quantum algebra. Given a 3-manifold M and a finite-dimensional Hopf algebra H, a Heegaard diagram for M can be read as a scalar-valued word in the structure tensors of H. Objects such as R-matrices, associators, or representation categories are not needed in the construction.

If time permits, I will discuss a converse result: The formal word problem for finite, involutory Hopf objects can be solved using 3-manifold topology.