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  "body": "\n## Statement\n\nLet $M$ be a closed connected [[orientable|702SNFQG0W]] irreducible [[3-manifold|R20TB2KEQQ]], where irreducible means that every embedded $2$-sphere bounds a $3$-ball.\nA compact $3$-manifold $N$ is *atoroidal* if every [[incompressible|6N9GW98MR6]] torus in $N$ is parallel to a boundary torus of $N$.\nThen $M$ contains a finite collection $\\mathcal{T}$ of disjoint embedded incompressible tori with the following property.\nEach component of $M$ cut along $\\mathcal{T}$ is atoroidal or a [[Seifert fiber space|9V1GF5WMKA]].\nA minimal collection $\\mathcal{T}$ with this property is unique up to isotopy.\n\n## Remarks\n\nThe theorem is due to Jaco and Shalen, and independently to Johannson.\nExistence rests on Haken's finiteness theorem, which bounds the number of disjoint, pairwise non-parallel incompressible surfaces.\nUniqueness comes from the study of how incompressible tori meet Seifert fibered pieces.\nA version of the theorem holds for compact manifolds whose boundary is a union of tori.\nFor example, take a [[satellite knot|A1E8TDEVJ9]] in $S^3$.\nThe boundary of a neighborhood of the companion knot is an incompressible torus in the knot complement, so the complement is not atoroidal.\nIf $M$ is itself Seifert fibered or atoroidal, the minimal collection is empty.\n\n## Explanations\n\n### L1 human\n\nAfter splitting a $3$-manifold into prime pieces, one can split each piece further along essential tori.\nThe JSJ theorem says there is a best way to do this, and that it is unique.\nEach resulting piece is either built from circles in a regular way or contains no essential torus at all.\n\n### L2 human\n\nThe JSJ decomposition is the second step of geometrization, after the prime decomposition.\nBy the geometrization theorem, the atoroidal pieces that are not Seifert fibered have hyperbolic interiors.\nSo every closed orientable irreducible $3$-manifold is assembled from Seifert fibered and hyperbolic pieces glued along tori.\n",
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