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The colored Jones polynomial of the figure-eight knot and a quantum modularity

Published online by Cambridge University Press:  20 March 2023

Hitoshi Murakami Open the ORCID record for Hitoshi Murakami [Opens in a new window]
Hitoshi Murakami*
Affiliation:
Graduate School of Information Sciences, Tohoku University, Aramaki-aza-Aoba 6-3-09, Aoba-ku, Sendai 980-8579, Japan
*
e-mail hitoshi@tohoku.ac.jp
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Abstract

We study the asymptotic behavior of the N-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp \bigl ((u+2p\pi \sqrt {-1})/N\bigr )$, where u is a small real number and p is a positive integer. We show that it is asymptotically equivalent to the product of the p-dimensional colored Jones polynomial evaluated at $\exp \bigl (4N\pi ^2/(u+2p\pi \sqrt {-1})\bigr )$ and a term that grows exponentially with growth rate determined by the Chern–Simons invariant. This indicates a quantum modularity of the colored Jones polynomial.

Keywords

Colored Jones polynomialvolume conjecturefigure-eight knotChern–Simons invariantReidemeister torsionquantum modularity

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Type
Article
Information
Canadian Journal of Mathematics , Volume 76 , Issue 2 , April 2024 , pp. 519 - 554
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work was supported by JSPS KAKENHI Grant Numbers JP22H01117, JP20K03601, and JP20K03931.

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