On the quandles of isometries of the hyperbolic 3-space
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914829427539968 |
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| author | Kai, Ryoya |
| author_facet | Kai, Ryoya |
| contents | A quandle is an algebraic structure whose axioms are related to the Reidemeister moves used in knot theory. In this paper, we investigate the conjugate quandle of the orientation-preserving isometry group $\mathrm{PSL}(2, \mathbb{C})$ of hyperbolic 3-space and its subquandles. We introduce a quandle, denoted by $Q(Γ, γ)$, associated with a pair $(Γ, γ)$. Here, $Γ$ is a Kleinian group, and $γ$ is a non-trivial element of $Γ$. This construction can be regarded as a generalization of knot quandles to hyperbolic knots. Moreover, for pairs $(Γ, γ)$ satisfying certain conditions, we construct the canonical map from $Q(Γ, γ)$ to the conjugate quandle of $\mathrm{PSL}(2, \mathbb{C})$, which is an injective quandle homomorphism with a discrete image. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04715 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the quandles of isometries of the hyperbolic 3-space Kai, Ryoya Geometric Topology A quandle is an algebraic structure whose axioms are related to the Reidemeister moves used in knot theory. In this paper, we investigate the conjugate quandle of the orientation-preserving isometry group $\mathrm{PSL}(2, \mathbb{C})$ of hyperbolic 3-space and its subquandles. We introduce a quandle, denoted by $Q(Γ, γ)$, associated with a pair $(Γ, γ)$. Here, $Γ$ is a Kleinian group, and $γ$ is a non-trivial element of $Γ$. This construction can be regarded as a generalization of knot quandles to hyperbolic knots. Moreover, for pairs $(Γ, γ)$ satisfying certain conditions, we construct the canonical map from $Q(Γ, γ)$ to the conjugate quandle of $\mathrm{PSL}(2, \mathbb{C})$, which is an injective quandle homomorphism with a discrete image. |
| title | On the quandles of isometries of the hyperbolic 3-space |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2406.04715 |