A complete invariant for closed surfaces in the three-sphere

TL;DR

This paper introduces a new complete invariant for closed surfaces in the 3-sphere, enabling classification of related topological structures through a constructed fundamental group diagram.

Contribution

It develops a complete invariant based on fundamental group diagrams for surfaces in the 3-sphere, extending existing theorems and providing computable invariants for handlebody links.

Findings

Proves the diagram is a complete invariant for embedded surfaces in the 3-sphere.

Extends the Kneser conjecture to 3-manifolds with boundary.

Calculates invariants for handlebody links.

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.