Incompressibility, Seifert fibrations, and JSJ systems

Let $M$ be a compact smooth $3$-manifold. A properly embedded surface is two-sided if it has a product neighbourhood $F\times [-1,1]$ in $M$. A compressing disk for a two-sided surface $F$ is an embedded disk $D\subset M$ with $\mathrm{int}(D)\cap F=\mathrm{\varnothing }$ and $\mathrm{\partial }D$ an essential simple closed curve in $F$; $F$ is incompressible if it has no compressing disk. A boundary-compressing disk is an embedded disk $D$ with $\mathrm{\partial }D=\alpha \cup \beta$, where $\alpha =D\cap F$ is an arc not cutting off a disk in $F$, $\beta =D\cap \mathrm{\partial }M$, and $\mathrm{int}(D)$ misses $F\cup \mathrm{\partial }M$; $F$ is boundary-incompressible if it has no such disk. A surface is boundary-parallel if it cobounds a product with a subsurface of $\mathrm{\partial }M$, and it is essential if it is incompressible, boundary-incompressible, and not boundary-parallel. $M$ is irreducible if every embedded $2$-sphere in its interior bounds an embedded $3$-ball in $M$. An incompressible torus is boundary-parallel if it cobounds a product $T^2\times [0,1]$ with a component of $\mathrm{\partial }M$; $M$ is atoroidal if every incompressible torus is boundary-parallel. A Seifert fibration is a decomposition into circles with local models given by fibered solid tori; in the model $(D^2\times S^1)\mathrm{/}⟨(z,w)\mapsto (e^{2\pi i\mathrm{/}q}z,e^{2\pi ip\mathrm{/}q}w)⟩$, where $\gcd (p,q)=1$, the core fiber has multiplicity $q$. Fibers of multiplicity $1$ are regular and those of multiplicity greater than $1$ are exceptional. A finite family of pairwise disjoint incompressible tori is a JSJ system if every component obtained by cutting $M$ along the family is atoroidal or Seifert-fibered. It is inclusion-minimal if no proper subfamily is a JSJ system. Atoroidal does not mean hyperbolic.

Allen Hatcher, Notes on Basic 3-Manifold Topology

Walter Neumann and Gadde Swarup, Canonical decompositions of 3-manifolds