Invariants from involutory Hopf algebras of 3-manifolds with embedded framed links

Abstract

We give invariants of pairs $(M,L)$ consisting of a closed connected oriented three-manifold and an (oriented) framed link $L$ embedded in $M$. This invariant generalizes the Kuperberg and Hennings-Kauffman-Radford (HKR) invariants of three-manifolds. We define Heegaard-Link diagrams which represent the pair $(M,L)$ and use the data of an involutory Hopf algebra and a representation of the Drinfeld double to construct the invariant. We show that if $L$ is the empty link, then the invariant recovers the Kuperberg invariant, and if $M$ is the three-sphere and certain particular representation is chosen, then the invariant recovers the HKR invariant. We also show that if the representation is the left regular representation of the Drinfeld double, then we recover the Kuperberg invariant of the surgery manifold $M(L)$, contributing to a new proof of the relationship between the HKR and Kuperberg invariants in the semisimple setting. To this end, we give a Heegaard diagram for $M(L)$ coming from the Heegaard-Link diagram representing the pair $(M,L)$. We also introduce a colored link invariant extending the construction and show it recovers the Witten-Reshetikhin-Turaev colored link invariant.