Hopf algebras and topological invariants of 3-manifolds

Topological invariants of knots and 3-manifolds can be divided into two classes, namely, classical invariants (those coming from algebraic topology) and quantum invariants (those defined using the representation theory of certain Hopf algebras.)

 

In this talk, I will discuss an approach to quantum invariants of closed 3-manifolds due to G. Kuperberg that relies on the Hopf algebras themselves instead of their representation theory. I will explain how this approach extends to the much wider class of sutured 3-manifolds, which includes links and Seifert surface complements, and how it recovers a powerful classical invariant when specialised to an exterior Hopf algebra, namely, the Reidemeister torsion and its twisted versions.