A preorder on the set of links with applications to symmetric unions
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| Format: | Preprint |
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2025
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| author | Boileau, Michel Kitano, Teruaki Nozaki, Yuta |
| author_facet | Boileau, Michel Kitano, Teruaki Nozaki, Yuta |
| contents | For a link $L$ in the $3$-sphere, the $π$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group $G(L)$ of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$ fitting into a certain commutative diagram, we define a relation $L\succeq L'$ and explore the relationships between the two links. Specifically, we prove that if $L\succeq L'$ and $L$ is a Montesinos link with $r$ rational tangles $(r\geq 3)$, then $L'$ is either a Montesinos link with at most $r+1$ rational tangles or a certain connected sum. We further show that if $L$ is a small link, then there are only finitely many links $L'$ satisfying $L\succeq L'$. In contrast, if $L$ has determinant zero, then $L\succeq L'$ for every $2$-bridge link $L'$. Our main applications concern symmetric unions of knots. In particular, we provide a criterion showing that a given knot does not admit a symmetric union presentation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12372 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A preorder on the set of links with applications to symmetric unions Boileau, Michel Kitano, Teruaki Nozaki, Yuta Geometric Topology 57K10, 57R18 (Primary) 57K32, 57M12 (Secondary) For a link $L$ in the $3$-sphere, the $π$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group $G(L)$ of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$ fitting into a certain commutative diagram, we define a relation $L\succeq L'$ and explore the relationships between the two links. Specifically, we prove that if $L\succeq L'$ and $L$ is a Montesinos link with $r$ rational tangles $(r\geq 3)$, then $L'$ is either a Montesinos link with at most $r+1$ rational tangles or a certain connected sum. We further show that if $L$ is a small link, then there are only finitely many links $L'$ satisfying $L\succeq L'$. In contrast, if $L$ has determinant zero, then $L\succeq L'$ for every $2$-bridge link $L'$. Our main applications concern symmetric unions of knots. In particular, we provide a criterion showing that a given knot does not admit a symmetric union presentation. |
| title | A preorder on the set of links with applications to symmetric unions |
| topic | Geometric Topology 57K10, 57R18 (Primary) 57K32, 57M12 (Secondary) |
| url | https://arxiv.org/abs/2510.12372 |