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We prove a 20-year-old conjecture concerning two quantum invariants of threemanifolds that are constructed from finite dimensional Hopf algebras, namely,the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The twoinvariants can be viewed as a non-semisimple generalization of theTuraev-Viro-Barrett-Westbury $(\text{TVBW})$ invariant and theWitten-Reshetikhin-Turaev $(\text{WRT})$ invariant, respectively. By aclassical result relating $\text{TVBW}$ and $\text{WRT}$, it follows that theKuperberg invariant for a semisimple Hopf algebra is equal to theHennings-Kauffman-Radford invariant for the Drinfeld double of the Hopfalgebra. However, whether the relation holds for non-semisimple Hopf algebrashas remained open, partly because the introduction of framings in this casemakes the Kuperberg invariant significantly more complicated to handle. We givean affirmative answer to this question. An important ingredient in the proofinvolves using a special Heegaard diagram in which one family of circles givesthe surgery link of the three manifold represented by the Heegaard diagram.

On Two Invariants of Three Manifolds from Hopf Algebras