Torsion in Kauffman bracket skein module of a $4$-strand Montesinos knot exterior
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917164366168064 |
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| author | Chen, Haimiao |
| author_facet | Chen, Haimiao |
| contents | For an oriented $3$-manifold $M$, let $\mathcal{S}(M)$ denote its Kauffman bracket skein module over $\mathbb{Z}[q^{\pm\frac{1}{2}}]$. We show that $\mathcal{S}(M)$ admits torsion when $M$ is the exterior of the Montesinos knot $K(a_1/b_1,a_2/b_2,a_3/b_4,a_4/b_4)$ with each $b_i\ge 3$. This provides a negative answer to Problem 1.92 (G)-(i) in the Kirby's list, which asks whether $\mathcal{S}(M)$ is free when $M$ is irreducible and has no incompressible non-boundary parallel torus. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_01177 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Torsion in Kauffman bracket skein module of a $4$-strand Montesinos knot exterior Chen, Haimiao Geometric Topology 57K31, 57K16 For an oriented $3$-manifold $M$, let $\mathcal{S}(M)$ denote its Kauffman bracket skein module over $\mathbb{Z}[q^{\pm\frac{1}{2}}]$. We show that $\mathcal{S}(M)$ admits torsion when $M$ is the exterior of the Montesinos knot $K(a_1/b_1,a_2/b_2,a_3/b_4,a_4/b_4)$ with each $b_i\ge 3$. This provides a negative answer to Problem 1.92 (G)-(i) in the Kirby's list, which asks whether $\mathcal{S}(M)$ is free when $M$ is irreducible and has no incompressible non-boundary parallel torus. |
| title | Torsion in Kauffman bracket skein module of a $4$-strand Montesinos knot exterior |
| topic | Geometric Topology 57K31, 57K16 |
| url | https://arxiv.org/abs/2311.01177 |