On a canonical placement of knots in irreducible 3-manifolds (Q1600213)

Let \(M\) be a compact connected orientable irreducible 3-manifold. A 2-submanifold \(T\) of \(M\) is a system of Jaco-Shalen-Johannson if (i) all components of \(T\) are incompressible tori and (ii) if each component of \(cl(M-N(T))\) (where \(N(T)\) is a closed regular neighborhood of \(T\)) either admits a Seifert fibration, or is atoroidal or is a torus bundle over the circle. If no subsystem of \(T\) has the same property, then \(T\) is a minimal system of Jaco-Shalen-Johannson. A knot type \(K\) in \(M\) is called global if no representative of it is contained in an embedded ball of \(M\). \(K\) in \(M\) is called isolable if it admits a representative disjoint from a minimal system of Jaco-Shalen-Johannson. Given a minimal system of Jaco-Shalen-Johannson \(T\) in \(M\), the pieces of \(M\) are the regular neighborhoods of incompressible tori in \(T \cup \partial M\), the components of the complement of these neighborhoods or regular neighborhoods of Seifert fibres in those components that admit Seifert fibrations. Let \(K\) be a global and isolable knot type in \(M\). The main result of the article describes the set of pieces of \(M\) that can contain a representative of \(K\). Furthermore the isotopy class of a representative of \(K\) inside a piece \(P\) is independent of the chosen representative.