Octahedral coordinates from the Wirtinger presentation
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916743223443456 |
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| author | McPhail-Snyder, Calvin |
| author_facet | McPhail-Snyder, Calvin |
| contents | Let $ρ$ be a representation of a knot group (or more generally, the fundamental group of a tangle complement) into $\operatorname{SL}_2(\mathbb{C})$ expressed in terms of the Wirtinger generators of a diagram $D$. This diagram also determines an ideal triangulation of the complement called the octahedral decomposition. $ρ$ induces a hyperbolic structure on the complement of $D$, and in this note we give a direct algebraic formula for the geometric parameters of the octahedral decomposition induced by this structure. Our formula gives a new, explicit criterion for whether $ρ$ occurs as a critical point of the diagram's Neumann-Zagier--Yokota potential function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19155 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Octahedral coordinates from the Wirtinger presentation McPhail-Snyder, Calvin Geometric Topology 57K32 (Primary), 57M05 (Secondary) Let $ρ$ be a representation of a knot group (or more generally, the fundamental group of a tangle complement) into $\operatorname{SL}_2(\mathbb{C})$ expressed in terms of the Wirtinger generators of a diagram $D$. This diagram also determines an ideal triangulation of the complement called the octahedral decomposition. $ρ$ induces a hyperbolic structure on the complement of $D$, and in this note we give a direct algebraic formula for the geometric parameters of the octahedral decomposition induced by this structure. Our formula gives a new, explicit criterion for whether $ρ$ occurs as a critical point of the diagram's Neumann-Zagier--Yokota potential function. |
| title | Octahedral coordinates from the Wirtinger presentation |
| topic | Geometric Topology 57K32 (Primary), 57M05 (Secondary) |
| url | https://arxiv.org/abs/2404.19155 |