Structure and unique factorization in concordance groups of links
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917389759676416 |
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| author | Sato, Kouki Yasuhara, Akira |
| author_facet | Sato, Kouki Yasuhara, Akira |
| contents | Donald and Owens introduced two link concordance groups with a marked component and showed that they contain the knot concordance group as a direct summand with infinitely generated complements. While not explicitly posed by Donald and Owens, the problem of determining the structure of these complements arises naturally from their work. In this paper, we completely resolve this problem by proving that both complements are isomorphic to $\mathbb{Z}^{\infty} \oplus (\mathbb{Z}/2\mathbb{Z})^{\infty}$. Moreover, we introduce a notion of prime element and establish a unique prime decomposition theorem. This yields a canonical normal form, providing a complete description of the group structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06514 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structure and unique factorization in concordance groups of links Sato, Kouki Yasuhara, Akira Geometric Topology 57K10 Donald and Owens introduced two link concordance groups with a marked component and showed that they contain the knot concordance group as a direct summand with infinitely generated complements. While not explicitly posed by Donald and Owens, the problem of determining the structure of these complements arises naturally from their work. In this paper, we completely resolve this problem by proving that both complements are isomorphic to $\mathbb{Z}^{\infty} \oplus (\mathbb{Z}/2\mathbb{Z})^{\infty}$. Moreover, we introduce a notion of prime element and establish a unique prime decomposition theorem. This yields a canonical normal form, providing a complete description of the group structure. |
| title | Structure and unique factorization in concordance groups of links |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2604.06514 |