On the gauge invariance of the Kuperberg invariant of certain high genus framed 3-manifolds

We show that the Kuperberg invariant of the Weeks manifold with any framing is a gauge invariant of finite-dimensional Hopf algebras, which provides the first example of gauge invariants of general finite-dimensional Hopf algebras via hyperbolic 3-manifolds. We also show that the Kuperberg invariant of the 3-torus is gauge invariant, which further supports the idea of systematically producing gauge invariants of Hopf algebras via topological methods proposed in \cite{CNW25}.