Character Varieties of Generalized Torus Knot Groups
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914032723689472 |
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| author | Florentino, Carlos Lawton, Sean |
| author_facet | Florentino, Carlos Lawton, Sean |
| contents | Given $\mathbf{n}=(n_{1},\ldots,n_{r})\in\mathbb{N}^r$, let $Γ_{\mathbf{n}}$ be a group presentable as $$\left\langle γ_{1},\ldots,γ_{r}\:|\:γ_{1}^{n_{1}}=γ_{2}^{n_{2}}=\cdots=γ_{r}^{n_{r}}\right\rangle. $$ If $\gcd(n_i,n_j)=1$ for all $i\not=j$, we say $Γ_{\mathbf{n}}$ is a {\it generalized torus knot group} and otherwise say it is a {\it generalized torus link group}. This definition includes torus knot and link groups ($r=2$), that is, fundamental groups of the complement of a torus knot or link in $S^{3}$. Let $G$ be a connected complex reductive affine algebraic group. We show that the $G$-character varieties of generalized torus knot groups are path-connected. We then count the number of irreducible components of the $\mathrm{SL}(2,\mathbb{C})$-character varieties of $Γ_{\mathbf{n}}$ when $n_i$ is odd for all $i$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15228 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Character Varieties of Generalized Torus Knot Groups Florentino, Carlos Lawton, Sean Geometric Topology Algebraic Geometry Group Theory Representation Theory Primary 14M35, 20F38, 14P25, Secondary 14L30, 14D20, 22E46 Given $\mathbf{n}=(n_{1},\ldots,n_{r})\in\mathbb{N}^r$, let $Γ_{\mathbf{n}}$ be a group presentable as $$\left\langle γ_{1},\ldots,γ_{r}\:|\:γ_{1}^{n_{1}}=γ_{2}^{n_{2}}=\cdots=γ_{r}^{n_{r}}\right\rangle. $$ If $\gcd(n_i,n_j)=1$ for all $i\not=j$, we say $Γ_{\mathbf{n}}$ is a {\it generalized torus knot group} and otherwise say it is a {\it generalized torus link group}. This definition includes torus knot and link groups ($r=2$), that is, fundamental groups of the complement of a torus knot or link in $S^{3}$. Let $G$ be a connected complex reductive affine algebraic group. We show that the $G$-character varieties of generalized torus knot groups are path-connected. We then count the number of irreducible components of the $\mathrm{SL}(2,\mathbb{C})$-character varieties of $Γ_{\mathbf{n}}$ when $n_i$ is odd for all $i$. |
| title | Character Varieties of Generalized Torus Knot Groups |
| topic | Geometric Topology Algebraic Geometry Group Theory Representation Theory Primary 14M35, 20F38, 14P25, Secondary 14L30, 14D20, 22E46 |
| url | https://arxiv.org/abs/2401.15228 |