A complete invariant for closed surfaces in the three-sphere

In this paper we use diagrams in categories to construct a complete invariant, the fundamental tree, for closed surfaces in the (based) $3$-sphere, which generalizes the knot group and its peripheral system. From the fundamental tree, we derive some computable invariants that are capable to distinguish inequivalent handlebody links with homeomorphic complements. To prove the completeness of the fundamental tree, we generalize the Kneser conjecture to $3$-manifolds with boundary, a topic interesting in its own right.