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The colored Jones polynomial of the figure-eight knot and a quantum modularity
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 2 / April 2024
- Published online by Cambridge University Press:
- 20 March 2023, pp. 519-554
- Print publication:
- April 2024
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- Article
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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.
