dorsal/arxiv
View SchemaGenealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family
| Authors | Thomas Kerler |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9601021 |
| URL | https://arxiv.org/abs/q-alg/9601021 |
Abstract
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 algebra, $\A$, that allows us to apply the Hennings algorithm directly, in order to compute $\tau_{RT}$, which is originally obtained from the semisimple trace-subquotient of $\A-mod$. Moreover, we generalize Hennings' rules to the context of cobordisms, in order to obtain a TQFT for connected surfaces compatible with $\tau_{HKR}\,$. As an application we show that, for lens spaces and $\A=U_q(sl_2)\,$, the ratio of $\tau_{HKR}$ and $\tau_{RT}$ is the order of the first homology group. In the course of this paper we also outline the topology and the algebra that enter invariance proofs, which contain no reference to 2-handle slides, but to other moves that are local. Finally, we give a list of open questions regarding cellular invariants, as defined by Turaev-Viro, Kuperberg, and others, their relations among each other, and their relations to the surgical invariants from above.
{
"annotation_id": "54b42f43-4b76-4210-ba88-7c84ce948c01",
"date_created": "2026-03-02T18:01:27.545000Z",
"date_modified": "2026-03-02T18:01:27.545000Z",
"file_hash": "cb8b32c626b24160afd95f0c413f853ea1e7949fe4c888ee3f1714dced586621",
"private": false,
"record": {
"abstract": "We study the relations between the invariants $\\tau_{RT}$, $\\tau_{HKR}$, and\n $\\tau_L$ of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko,\n respectively. In particular, we discuss explicitly how $\\tau_L$ specializes\nto $\\tau_{RT}$ for semisimple categories and to $\\tau_{HKR}$ for Tannakian\ncategories. We give arguments for that $\\tau_L$ is the most general invariant\nthat stems from an extended TQFT. We introduce a canonical, central element,\n{\\sf Q}, for a quasi-triangular Hopf algebra, $\\A$, that allows us to apply the\nHennings algorithm directly, in order to compute $\\tau_{RT}$, which is\noriginally obtained from the semisimple trace-subquotient of $\\A-mod$.\nMoreover, we generalize Hennings\u0027 rules to the context of cobordisms, in order\nto obtain a TQFT for connected surfaces compatible with $\\tau_{HKR}\\,$. As an\napplication we show that, for lens spaces and $\\A=U_q(sl_2)\\,$, the ratio of\n$\\tau_{HKR}$ and $\\tau_{RT}$ is the order of the first homology group. In the\ncourse of this paper we also outline the topology and the algebra that enter\ninvariance proofs, which contain no reference to 2-handle slides, but to other\nmoves that are local. Finally, we give a list of open questions regarding\ncellular invariants, as defined by Turaev-Viro, Kuperberg, and others, their\nrelations among each other, and their relations to the surgical invariants from\nabove.",
"arxiv_id": "q-alg/9601021",
"authors": [
"Thomas Kerler"
],
"categories": [
"q-alg",
"hep-th",
"math.QA"
],
"title": "Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family",
"url": "https://arxiv.org/abs/q-alg/9601021"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "bd68d8ab-e6b0-41e0-97f8-4949883a29d2",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}