I would like to discuss three-manifold invariants built from link-invariants using surgery representations of three-manifolds. This is the basis of the Reshetikhin-Turaev construction, which produces invariants that behave nicely under cutting and glueing, specifically, they define three-dimensional topological field theories. The algebraic data needed to describe these theories are so-called modular tensor categories, as provided for example by representations of certain Hopf algebras. While the original construction is limited to the semisimple case, in recent developments also non-semisimple modular categories can be used to define topological field theories, leading to some effects not present in the semisimple situation.
