An extended symmetric union and its Alexander polynomial
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910822806061056 |
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| author | Kitano, Teruaki Nakae, Yasuharu |
| author_facet | Kitano, Teruaki Nakae, Yasuharu |
| contents | For prime knots $K_1$ and $K_2$, we write $K_1 \geq K_2$ if there is an epimorphism from the knot group of $K_1$ to that of $K_2$ which preserves the meridian. We construct a family of pairs of knots with $K_1 \geq K_2$ such that an epimorphism maps the longitude of $K_1$ to the trivial element. This construction is regarded as an extension of a symmetric union with a single full twisted region. In particular, it extends a property of the Alexander polynomial of a symmetric union. We also exhibit that all but two of the knots up to ten crossings in the list of Kitano-Suzuki, which have an epimorphism mapping the longitude to the trivial element, arise from this construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An extended symmetric union and its Alexander polynomial Kitano, Teruaki Nakae, Yasuharu Geometric Topology Primary 57K10, Secondary 57M05 For prime knots $K_1$ and $K_2$, we write $K_1 \geq K_2$ if there is an epimorphism from the knot group of $K_1$ to that of $K_2$ which preserves the meridian. We construct a family of pairs of knots with $K_1 \geq K_2$ such that an epimorphism maps the longitude of $K_1$ to the trivial element. This construction is regarded as an extension of a symmetric union with a single full twisted region. In particular, it extends a property of the Alexander polynomial of a symmetric union. We also exhibit that all but two of the knots up to ten crossings in the list of Kitano-Suzuki, which have an epimorphism mapping the longitude to the trivial element, arise from this construction. |
| title | An extended symmetric union and its Alexander polynomial |
| topic | Geometric Topology Primary 57K10, Secondary 57M05 |
| url | https://arxiv.org/abs/2502.08229 |