A complete invariant for closed surfaces in the three-sphere

Abstract

Associated to an embedded surface in the $3$-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, wherefrom we deduce complete invariants of handlebody links, tunnels of handlebody links, and spatial graphs.The main ingredients in the proof of the completeness are a generalization of the Kneser conjecture for $3$-manifolds with boundary proved also here, and extensions of Waldhausen's theorem by Evans, Tucker and Swarup. Computable invariants of handlebody links derived therefrom are calculated.

20 pages, 6 figures, proof of the main theorem simplified, following referee's report; section 6 restructured: Theorem 1.4 in v1 removed due to a gap in its proof; a new example added