具有指定纽结作为分支的零亏格链环
Genus-zero links with prescribed knots as components
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中文总结 AI 辅助
研究三维流形中纽结同痕类成为零亏格链环分支的条件，通过满足共轭类要求实现，对\(M = \mathbb S^3\)还控制环绕数，用4 - 亏格代替3 - 亏格时有类似结果，仅指定两个纽结同痕类。
中文摘要 AI 辅助
我们证明，在三维流形\(M\)中，任意至少三个纽结同痕类的有限集合，只要满足它们在\(\pi_1(M)\)中共轭类的一个明显要求，就可实现为\(M\)中一个零亏格链环的分支。对于\(M = \mathbb S^3\)，此条件自动满足，且在此情形下我们还控制了分支的两两环绕数。用4 - 亏格代替3 - 亏格，我们得到一个类似结果，其中仅指定两个纽结同痕类。
英文摘要
We prove that any finite collection of at least three isotopy classes of knots in a 3-manifold $M$ is realizable as the components of a genus-zero link in $M$, provided that an obvious requirement on their conjugacy classes in $π_1(M)$ is met. This condition is vacuously satisfied for $M = \mathbb S^3$, and in this case we also control the pairwise linking numbers of the components. Replacing the 3-genus with the 4-genus, we obtain an analogous result where only two knot isotopy classes are prescribed.
