|Z_{Kup}|=|Z_{Henn}|^2 for Lens spaces

TL;DR

This paper proves a precise relationship between two quantum invariants of lens spaces, showing that the absolute value of Z_{Kup} equals the square of Z_{Henn}, for a specific class of Hopf algebra-based invariants.

Contribution

It establishes a fundamental equality between Z_{Kup} and Z_{Henn} invariants for lens spaces using factorizable finite dimensional ribbon Hopf algebras.

Findings

|Z_{Kup}|=|Z_{Henn}|^2 for lens spaces

Validates the relationship for factorizable finite dimensional ribbon Hopf algebras

Enhances understanding of quantum invariants in 3-manifold topology

Abstract

M. Hennings and G. Kuperberg defined quantum invariants Z_{Henn} and Z_{Kup} of closed oriented 3-manifolds based on certain Hopf algebras, respectively. We prove that |Z_{Kup}|=|Z_{Henn}|^2 for lens spaces when both invariants are based on factorizable finite dimensional ribbon Hopf algebras.