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| Main Authors: | , |
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| Format: | Preprint |
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2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1708.01511 |
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| _version_ | 1866909654657794048 |
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| author | Nagasato, Fumikazu Suzuki, Shinnosuke |
| author_facet | Nagasato, Fumikazu Suzuki, Shinnosuke |
| contents | We show that the (4,5)-torus knot $T_{4,5}$ admits exactly one ghost character. We then show that this ghost character provides the following two important results. (1) It is known that for any knot $K$ every (meridionally) trace-free $\SL_2(\C)$-representation of the knot group $G(K)$ yields an $\SL_2(\C)$-representation of the fundamental group $π_1(Σ_2K)$ of the 2-fold branched cover $Σ_2K$ of the 3-sphere along $K$. This correspondence often but not always provides all $\SL_2(\C)$-representations of $π_1(Σ_2K)$. We show by using the ghost character that $T_{4,5}$ is the simplest torus knot such that $π_1(Σ_2T_{4,5})$ admits an $\SL_2(\C)$-representation which cannot be realized by any trace-free $\SL_2(\C)$-representations. (2) We show that $T_{4,5}$ is the simplest torus knot that provides a counterexample to Ng's conjecture, concerned with a polynomial map $h^*$ between the character variety $X(Σ_2K)$ of $π_1(Σ_2K)$ and the fundamental variety $F_2(K)$. More precisely, the map $h^*$ is surjective but not injective, and hence not an isomorphism for $T_{4,5}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_01511 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | The ghost character of the (4,5)-torus knot and its applications Nagasato, Fumikazu Suzuki, Shinnosuke Geometric Topology 57M27 (Primary), 57M25 (Secondary) We show that the (4,5)-torus knot $T_{4,5}$ admits exactly one ghost character. We then show that this ghost character provides the following two important results. (1) It is known that for any knot $K$ every (meridionally) trace-free $\SL_2(\C)$-representation of the knot group $G(K)$ yields an $\SL_2(\C)$-representation of the fundamental group $π_1(Σ_2K)$ of the 2-fold branched cover $Σ_2K$ of the 3-sphere along $K$. This correspondence often but not always provides all $\SL_2(\C)$-representations of $π_1(Σ_2K)$. We show by using the ghost character that $T_{4,5}$ is the simplest torus knot such that $π_1(Σ_2T_{4,5})$ admits an $\SL_2(\C)$-representation which cannot be realized by any trace-free $\SL_2(\C)$-representations. (2) We show that $T_{4,5}$ is the simplest torus knot that provides a counterexample to Ng's conjecture, concerned with a polynomial map $h^*$ between the character variety $X(Σ_2K)$ of $π_1(Σ_2K)$ and the fundamental variety $F_2(K)$. More precisely, the map $h^*$ is surjective but not injective, and hence not an isomorphism for $T_{4,5}$. |
| title | The ghost character of the (4,5)-torus knot and its applications |
| topic | Geometric Topology 57M27 (Primary), 57M25 (Secondary) |
| url | https://arxiv.org/abs/1708.01511 |