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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.05570 |
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| _version_ | 1866912751276785664 |
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| author | Kitaeff, Edwin |
| author_facet | Kitaeff, Edwin |
| contents | Given a knot $K$ and a generic slope $r$, we study the Kauffman bracket skein module (KBSM) $S(E_K (r) , \mathbb{Q} (A))$ of the Dehn filling $E_K (r)$ of slope $r$ along $K$, assuming that the KBSM $S(E_K , \mathbb{Q} [A^{\pm 1}])$ of the exterior $E_K$ of $K$ is finitely generated over $S(\partial E_K ,\mathbb{Q} [A^{\pm 1}])$. As shown in a paper of Thang Lê, this condition is satisfied for $K$ a two-bridge knot. In this setting, we show that $\dim_{\mathbb{C}} (S_ζ(E_K (r))) = \dim_{\mathbb{Q} (A)} (S (E_K (r)))$ for almost all primitive roots of unity $ζ$ of order $2N$ with $N$ odd, and for almost all slopes $r$. When the character variety of a 3-manifold $M$ is finite, we also discuss the decomposition of $S_ζ(M)$ in terms of localized skein modules. In particular, the dimension of the localized skein modules at a non-central point is the multiplicity of this point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05570 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dimension of the skein module of a Dehn filling Kitaeff, Edwin Geometric Topology Given a knot $K$ and a generic slope $r$, we study the Kauffman bracket skein module (KBSM) $S(E_K (r) , \mathbb{Q} (A))$ of the Dehn filling $E_K (r)$ of slope $r$ along $K$, assuming that the KBSM $S(E_K , \mathbb{Q} [A^{\pm 1}])$ of the exterior $E_K$ of $K$ is finitely generated over $S(\partial E_K ,\mathbb{Q} [A^{\pm 1}])$. As shown in a paper of Thang Lê, this condition is satisfied for $K$ a two-bridge knot. In this setting, we show that $\dim_{\mathbb{C}} (S_ζ(E_K (r))) = \dim_{\mathbb{Q} (A)} (S (E_K (r)))$ for almost all primitive roots of unity $ζ$ of order $2N$ with $N$ odd, and for almost all slopes $r$. When the character variety of a 3-manifold $M$ is finite, we also discuss the decomposition of $S_ζ(M)$ in terms of localized skein modules. In particular, the dimension of the localized skein modules at a non-central point is the multiplicity of this point. |
| title | Dimension of the skein module of a Dehn filling |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2512.05570 |