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}.

💡 Research Summary

The paper investigates the gauge invariance of the Kuperberg invariant associated with finite‑dimensional Hopf algebras when evaluated on two specific 3‑manifolds: the Weeks manifold (the closed hyperbolic 3‑manifold of smallest volume) and the 3‑torus. The authors aim to answer the question raised in earlier work (CNW25) whether every framed Kuperberg invariant is a gauge invariant, i.e., unchanged under Drinfeld twists (2‑cocycles) of the underlying Hopf algebra.

After a concise review of Hopf algebra basics—multiplication, comultiplication, antipode, integrals Λ and cointegrals λ, and the notion of a normalized pair (Λ, λ)—the paper recalls the intrinsic characterization of gauge equivalence via a normalized 2‑cocycle F. Twisting a Hopf algebra H by F yields a new Hopf algebra H_F with the same algebra structure but modified coproduct and antipode. The authors collect several technical identities (Proposition 1) concerning the twist element u, its inverse, the element Q = u S(u⁻¹), and the behavior of iterated coproducts under twisting. These identities are the algebraic backbone for proving invariance.

The Kuperberg invariant Z(M,f,H) for a framed closed 3‑manifold (M,f) is defined combinatorially using a Heegaard diagram. One chooses a genus‑g Heegaard splitting, draws lower curves η_i and upper curves μ_j, and equips the diagram with two admissible vector fields b₁ and b₂ (the third field b₃ follows from the right‑hand rule). For each intersection point p between a lower and an upper curve, rotation numbers θ_c(p) (tangent of the curve relative to b₁) and φ_c(p) (rotation of b₂ around b₁) are recorded. The quantities \