Associated to an embedded surface in the three-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, whereform we deduce complete invariants of handlebody links, tunnels of handlebody links, and spatial graphs. The main ingredients in the proof of the completeness include a generalization of the Kneser conjecture for three-manifolds with boundary proved here, and extensions of Waldhausen’s theorem by Evans, Tucker and Swarup. Computable invariants of handlebody links derived therefrom are calculated.

中文翻译：

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三球封闭曲面的完全不变量

结合三球内的嵌入曲面，我们构造了一个基本群图，并证明它是一个完全不变量，由此我们推导出把手体链接、把手体链接隧道和空间图的完全不变量。完整性证明的主要内容包括对此处证明的边界三流形的 Kneser 猜想的推广，以及 Evans、Tucker 和 Swarup 对 Waldhausen 定理的扩展。计算由此导出的手柄体链接的可计算不变量。