The generalized volume conjecture for the figure-eight knot parametrized by a complex number with small imaginary part
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912866758557696 |
|---|---|
| author | Murakami, Hitoshi |
| author_facet | Murakami, Hitoshi |
| contents | We study the asymptotic behavior, as $N$ tends to infinity, of the $N$-dimensional colored Jones polynomial of the figure-eight knot, evaluated at $\exp(ξ/N)$ for a complex parameter $ξ$ with $0<\mathrm{Im}ξ<π/2$. We prove that if $\mathrm{Re}ξ$ is large the colored Jones polynomial grows exponentially with growth rate expressed by the Chern--Simons invariant, and that if $\mathrm{Re}ξ$ is small it converges to the reciprocal of the Alexander polynomial evaluated at $\expξ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01049 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The generalized volume conjecture for the figure-eight knot parametrized by a complex number with small imaginary part Murakami, Hitoshi Geometric Topology We study the asymptotic behavior, as $N$ tends to infinity, of the $N$-dimensional colored Jones polynomial of the figure-eight knot, evaluated at $\exp(ξ/N)$ for a complex parameter $ξ$ with $0<\mathrm{Im}ξ<π/2$. We prove that if $\mathrm{Re}ξ$ is large the colored Jones polynomial grows exponentially with growth rate expressed by the Chern--Simons invariant, and that if $\mathrm{Re}ξ$ is small it converges to the reciprocal of the Alexander polynomial evaluated at $\expξ$. |
| title | The generalized volume conjecture for the figure-eight knot parametrized by a complex number with small imaginary part |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2602.01049 |