# Genealogy of Non-perturbative Quantum-Invariants of 3-Manifolds: The Surgical Family

@article{Kerler1995GenealogyON, title={Genealogy of Non-perturbative Quantum-Invariants of 3-Manifolds: The Surgical Family}, author={Thomas Kerler}, journal={arXiv: Quantum Algebra}, year={1995} }

We study the relations between the invariants $\tau_{RT}$, $\tau_{HKR}$, and
$\tau_L$ of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko,
respectively. In particular, we discuss explicitly how $\tau_L$ specializes to $\tau_{RT}$ for semisimple categories and to $\tau_{HKR}$ for Tannakian categories. We give arguments for that $\tau_L$ is the most general invariant that stems from an extended TQFT. We introduce a canonical, central element, {\sf Q}, for a quasi-triangular Hopf…

## 61 Citations

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We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The…

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Using an extension of the Kontsevich integral to tangles in handlebodies
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We introduce a family of factorisable ribbon quasi-Hopf algebras $Q(N)$ for $N$ a positive integer: as an algebra, $Q(N)$ is the semidirect product of $\mathbb{C}\mathbb{Z}_2$ with the direct sum of…

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We show a simple relation between Witten–Reshetikhin–Turaev SU(2) invariant and the Hennings invariant associated with the restricted quantum $${{\mathfrak{sl}_{2}}}$$ . These invariants are defined…

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Abstract We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the…

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For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:(1) ${\mathcal{C}}$ always contains a simple projective…

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We define the notion of a Kirby element of a ribbon category $\mathcal{C}$ (not necessarily semisimple). Kirby elements lead to 3-manifold invariants. We characterize a class of Kirby elements, the…

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