Fibering of double twist knots via the adjoint hyperbolic torsion polynomial
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911112274903040 |
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| author | Tran, Anh T. |
| author_facet | Tran, Anh T. |
| contents | For a hyperbolic knot $K$ in $S^3$, the adjoint hyperbolic torsion polynomial $\mathcal T^{\mathrm{Ad}}_K(t) \in \mathbb C[t^{\pm 1}]$ is defined as a normalization of the twisted Alexander polynomial of $K$ associated with the $\mathrm{SL}_3(\mathbb C)$-representation obtained by composing the holonomy representation of $K$ with the adjoint action of $\mathrm{SL}_2(\mathbb C)$ on its Lie algebra $\mathfrak{sl}_2(\mathbb C)$. In this paper we consider the adjoint hyperbolic torsion polynomial for a two-parameter family of rational knots called double twist knots, and show that $\mathcal T^{\mathrm{Ad}}_K(t)$ determines the genus and fibering of this family by using algebraic integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14731 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fibering of double twist knots via the adjoint hyperbolic torsion polynomial Tran, Anh T. Geometric Topology Primary 57K31, Secondary 57K14, 57M05 For a hyperbolic knot $K$ in $S^3$, the adjoint hyperbolic torsion polynomial $\mathcal T^{\mathrm{Ad}}_K(t) \in \mathbb C[t^{\pm 1}]$ is defined as a normalization of the twisted Alexander polynomial of $K$ associated with the $\mathrm{SL}_3(\mathbb C)$-representation obtained by composing the holonomy representation of $K$ with the adjoint action of $\mathrm{SL}_2(\mathbb C)$ on its Lie algebra $\mathfrak{sl}_2(\mathbb C)$. In this paper we consider the adjoint hyperbolic torsion polynomial for a two-parameter family of rational knots called double twist knots, and show that $\mathcal T^{\mathrm{Ad}}_K(t)$ determines the genus and fibering of this family by using algebraic integers. |
| title | Fibering of double twist knots via the adjoint hyperbolic torsion polynomial |
| topic | Geometric Topology Primary 57K31, Secondary 57K14, 57M05 |
| url | https://arxiv.org/abs/2508.14731 |