An extended symmetric union with multiple tangle regions and its Alexander polynomial
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917334272180224 |
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| author | Kitano, Teruaki Nakae, Yasuharu |
| author_facet | Kitano, Teruaki Nakae, Yasuharu |
| contents | The authors recently introduced a new construction of a knot as an extended symmetric union of a knot with a single tangle region. In this paper, we generalize the construction to include multiple tangle regions. The constructed knot $K$ with a partial knot $\hat{K}$ and multiple tangle regions satisfies the following two properties: its Alexander polynomial is the product of the Alexander polynomials of the numerators of these tangles and the square of the Alexander polynomial of the partial knot $\hat{K}$, and there exists a surjective homomorphism from the knot group of $K$ to that of $\hat{K}$ which maps the longitude of $K$ to the trivial element. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02800 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An extended symmetric union with multiple tangle regions and its Alexander polynomial Kitano, Teruaki Nakae, Yasuharu Geometric Topology Primary 57K10, Secondary 57M05 The authors recently introduced a new construction of a knot as an extended symmetric union of a knot with a single tangle region. In this paper, we generalize the construction to include multiple tangle regions. The constructed knot $K$ with a partial knot $\hat{K}$ and multiple tangle regions satisfies the following two properties: its Alexander polynomial is the product of the Alexander polynomials of the numerators of these tangles and the square of the Alexander polynomial of the partial knot $\hat{K}$, and there exists a surjective homomorphism from the knot group of $K$ to that of $\hat{K}$ which maps the longitude of $K$ to the trivial element. |
| title | An extended symmetric union with multiple tangle regions and its Alexander polynomial |
| topic | Geometric Topology Primary 57K10, Secondary 57M05 |
| url | https://arxiv.org/abs/2601.02800 |