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

TL;DR

This paper demonstrates that the Kuperberg invariant for certain hyperbolic 3-manifolds, like the Weeks manifold, is a gauge invariant of finite-dimensional Hopf algebras, establishing a new link between topology and algebra.

Contribution

It provides the first example of gauge invariants of general finite-dimensional Hopf algebras derived from hyperbolic 3-manifolds, expanding the scope of topological invariants.

Findings

Kuperberg invariant of the Weeks manifold is gauge invariant.

Kuperberg invariant of the 3-torus is gauge invariant.

Supports systematic production of gauge invariants via topological methods.

Abstract

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}.