Geometric structures and $PSL_2(\mathbb{C})$ representations of knot groups from knot diagrams
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918023527399424 |
|---|---|
| author | Petersen, Kathleen L. Tsvietkova, Anastasiia |
| author_facet | Petersen, Kathleen L. Tsvietkova, Anastasiia |
| contents | We describe a new method of producing equations for the canonical component of representation variety of a knot group into $PSL_2(\mathbb{C})$. Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of the knot complement, and uses only a knot diagram satisfying a few mild restrictions. This gives a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. The algorithm yields an explicit description for the hyperbolic structures (complete or incomplete) that correspond to geometric representations of a hyperbolic knot. As an illustration, we give the formulas for the equations for the variety of closed alternating braids $(σ_1(σ_2)^{-1})^n$ that depend only on $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric structures and $PSL_2(\mathbb{C})$ representations of knot groups from knot diagrams Petersen, Kathleen L. Tsvietkova, Anastasiia Geometric Topology 57K10, 57K31, 57K35, 57K32 We describe a new method of producing equations for the canonical component of representation variety of a knot group into $PSL_2(\mathbb{C})$. Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of the knot complement, and uses only a knot diagram satisfying a few mild restrictions. This gives a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. The algorithm yields an explicit description for the hyperbolic structures (complete or incomplete) that correspond to geometric representations of a hyperbolic knot. As an illustration, we give the formulas for the equations for the variety of closed alternating braids $(σ_1(σ_2)^{-1})^n$ that depend only on $n$. |
| title | Geometric structures and $PSL_2(\mathbb{C})$ representations of knot groups from knot diagrams |
| topic | Geometric Topology 57K10, 57K31, 57K35, 57K32 |
| url | https://arxiv.org/abs/2505.11779 |