In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson.
An alternative version of the JSJ decomposition states: The submanifold Σ with the smallest number of boundary tori is called the characteristic submanifold of M; it is unique (up to isotopy).
Cutting the manifold along the tori bounding the characteristic submanifold is also sometimes called a JSJ decomposition, though it may have more tori than the standard JSJ decomposition.
However there is a subtle difference: if one of the tori in the JSJ decomposition is "non-separating", then the boundary of the characteristic submanifold has two parallel copies of it (and the region between them is a Seifert manifold isomorphic to the product of a torus and a unit interval).
The set of tori bounding the characteristic submanifold can be characterised as the unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that closure of each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.