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arXiv · 2307.07100

The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation

Hitoshi Murakami
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Abstract

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp\bigl((\kappa+2p\pi\i/N\bigr)$, where $\kappa:=\arccosh(3/2)$ and $p$ is a positive integer. We can prove that it grows exponentially with growth rate determined by the Chern--Simons invariant of an affine representation from the fundamental group of the knot complement to the Lie group $\SL(2;\C)$.

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Hitoshi Murakami. 2023-07-14. The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation. https://doi.org/10.2140/agt.2025.25.3523

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