ABSTRACT
We study the relations between the invariants τRT, τHKR, and τL of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko, respectively. In particular, we discuss explicitly how τL specializes to τRT for semi-simple categories and to τHKR for Tannakian categories. We argue that τL is the most general invariant that stems from an extended tqft. We introduce a canonical, central element, Q, for a quasi-triangular Hopf algebra, A, that allows us to apply the Hennings algorithm directly, in order to compute τRT, which is originally obtained from the semi-simple trace-subquotient of A-mod. Moreover, we generalize Hennings’ rules to the context of cobordisms, in order to obtain a tqft for connected surfaces compatible with τHKR. As an application we show that, for lens spaces and A = Uq (sl 2), the ratio of τHKR and τRT is the order of the first homology group. In the course of this paper we also outline the topology and the algebra that enter invariance proofs, which contain no reference to 2-handle slides, but to other moves that are local. Finally, we give a list of open questions regarding cellular invariants, as defined by Turaev-Viro, Kuperberg, and others, their mutual relations, and their relations to the surgical invariants from above.
