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Quantum trace map for 3-manifolds and a 'length conjecture'

Prarit Agarwal Queen Mary University of London, Mile End Road, London E1 4NS, UK; Elaitra Ltd Dongmin Gang Department of Physics and Astronomy & Center for Theoretical Physics, Seoul National University, 1 Gwanak-ro, Seoul 08826, Korea; Asia Pacific Center for Theoretical Physics (APCTP), Pohang 37673, Korea Sangmin Lee College of Liberal Studies, Seoul National University, Seoul 08826, Korea; Department of Physics and Astronomy & Center for Theoretical Physics, Seoul National University, 1 Gwanak-ro, Seoul 08826, Korea Mauricio Andrés Romo Jorquera Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

Geometric Analysis and Geometric Topology arXiv subject: High Energy Physics - Theory (hep-th) mathscidoc:2207.15001

arXiv, 2022.3
We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement S^3∖K. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives in a module generated by products of quantized edge parameters of the ideal triangulation modulo some equivalence relations determined by gluing equations. Combining the quantum map with a state-integral model of SL(2,C) Chern-Simons theory, one can define perturbative invariants of knot K in the knot complement whose leading part is determined by its complex hyperbolic length. We then conjecture that the perturbative invariants determine an asymptotic expansion of the Jones polynomial for a link composed of K and K. We propose the explicit quantum trace map for figure-eight knot complement and confirm the length conjecture up to the second order in the asymptotic expansion both numerically and analytically.
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@inproceedings{prarit2022quantum,
  title={Quantum trace map for 3-manifolds and a 'length conjecture'},
  author={Prarit Agarwal, Dongmin Gang, Sangmin Lee, and Mauricio Andrés Romo Jorquera},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220706144246780062546},
  booktitle={arXiv},
  year={2022},
}
Prarit Agarwal, Dongmin Gang, Sangmin Lee, and Mauricio Andrés Romo Jorquera. Quantum trace map for 3-manifolds and a 'length conjecture'. 2022. In arXiv. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220706144246780062546.
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