Irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either "atoroidal" or "Seifert-fibered."
Jaco-Shalen-Johannson Torus Decomposition
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Cite this as:
Weisstein, Eric W. "Jaco-Shalen-Johannson Torus Decomposition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Jaco-Shalen-JohannsonTorusDecomposition.html