The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918160333012992 |
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| author | Murakami, Hitoshi |
| author_facet | Murakami, Hitoshi |
| contents | We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp\bigl((κ+2p\piı/N\bigr)$, where $κ:=\arccosh(3/2)$ 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 $\SL(2;\C)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07100 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation Murakami, Hitoshi Geometric Topology We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp\bigl((κ+2p\piı/N\bigr)$, where $κ:=\arccosh(3/2)$ 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 $\SL(2;\C)$. |
| title | The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2307.07100 |