The colored Jones polynomial of the figure-eight knot and a quantum modularity
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916237936689152 |
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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 $\exp\bigl((u+2p\piı)/N\bigr)$, 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 $\exp\bigl(4Nπ^2/(u+2p\piı)\bigr)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07751 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The colored Jones polynomial of the figure-eight knot and a quantum modularity Murakami, Hitoshi Geometric Topology We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp\bigl((u+2p\piı)/N\bigr)$, 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 $\exp\bigl(4Nπ^2/(u+2p\piı)\bigr)$ 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. |
| title | The colored Jones polynomial of the figure-eight knot and a quantum modularity |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2209.07751 |