A complete invariant for closed surfaces in the three-sphere

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.