The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

中文翻译：

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闭合面辫的同位素与同胚

在封闭的可定向表面 Ʃ 中辫子的闭合是 Ʃ ×小号1. 我们按照表面辫群的代数将这种闭合表面辫分类为同位素和同胚（闭合球辫的同位素具有小的不确定性）。我们发现，在正属中，靠近同位素链的辫子当且仅当它们是共轭的，并且当且仅当它们在表面辫群模上的映射类群的外部作用的相同轨道上时才接近同胚链它的中心。