### Abstract

We consider the asymptotics of the Turaev–Viro and the Reshetikhin–Turaev invariants of a hyperbolic 3-manifold, evaluated at the root of unity exp(2π√-1/r) instead of the standard exp(2π√-1/r). We present evidence that, as r tends to ∞, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten’s Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin–Turaev invariants may have a geometric interpretation other than the original one via SU(2) Chern–Simons gauge theory.

Original language | English (US) |
---|---|

Pages (from-to) | 419-460 |

Number of pages | 42 |

Journal | Quantum Topology |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2018 |

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### Keywords

- Reshetikhin-Turaev invariants
- Turaev-Viro invariants
- Volume conjecture

### ASJC Scopus subject areas

- Mathematical Physics
- Geometry and Topology

### Cite this

*Quantum Topology*,

*9*(3), 419-460. https://doi.org/10.4171/QT/111

**Volume conjectures for the reshetikhin–turaev and the turaev–viro invariants.** / Chen, Qingtao; Yang, Tian.

Research output: Contribution to journal › Article

*Quantum Topology*, vol. 9, no. 3, pp. 419-460. https://doi.org/10.4171/QT/111

}

TY - JOUR

T1 - Volume conjectures for the reshetikhin–turaev and the turaev–viro invariants

AU - Chen, Qingtao

AU - Yang, Tian

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We consider the asymptotics of the Turaev–Viro and the Reshetikhin–Turaev invariants of a hyperbolic 3-manifold, evaluated at the root of unity exp(2π√-1/r) instead of the standard exp(2π√-1/r). We present evidence that, as r tends to ∞, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten’s Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin–Turaev invariants may have a geometric interpretation other than the original one via SU(2) Chern–Simons gauge theory.

AB - We consider the asymptotics of the Turaev–Viro and the Reshetikhin–Turaev invariants of a hyperbolic 3-manifold, evaluated at the root of unity exp(2π√-1/r) instead of the standard exp(2π√-1/r). We present evidence that, as r tends to ∞, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten’s Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin–Turaev invariants may have a geometric interpretation other than the original one via SU(2) Chern–Simons gauge theory.

KW - Reshetikhin-Turaev invariants

KW - Turaev-Viro invariants

KW - Volume conjecture

UR - http://www.scopus.com/inward/record.url?scp=85050025222&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050025222&partnerID=8YFLogxK

U2 - 10.4171/QT/111

DO - 10.4171/QT/111

M3 - Article

AN - SCOPUS:85050025222

VL - 9

SP - 419

EP - 460

JO - Quantum Topology

JF - Quantum Topology

SN - 1663-487X

IS - 3

ER -