The JSJ system detects a companion torus when it is not amalgamated

Assume full AC. Let $K\subset S^3$ be a nontrivial smooth oriented knot with a specified orientation-preserving companion embedding $e:V=S^1\times D^2\to S^3$ that sends the oriented core to $K$ and identifies the product meridian and longitude with the positive meridian and preferred longitude of $K$. Let $P\subset \mathrm{int}V$ be a smooth oriented pattern with a chosen closed tubular neighbourhood $N(P)\subset \mathrm{int}V$. Put $X_P=V\setminus \mathrm{int}N(P)$ and assume that no properly embedded disk in $X_P$ has essential boundary on $\mathrm{\partial }V$, and that $X_P$ is not a product $T^2\times [0,1]$ with ends $\mathrm{\partial }V$ and $\mathrm{\partial }N(P)$. Put $P(K)=e(P)$, $M=E_{P(K)}=S^3\setminus \mathrm{int}e(N(P))$, and $T_c=e(\mathrm{\partial }V)$. Suppose both components of $M$ cut along $T_c$ are atoroidal or Seifert-fibered, and $M$ itself is neither atoroidal nor Seifert-fibered. Then every inclusion-minimal JSJ system of $M$ consists of one torus ambient isotopic to $T_c$. Cutting along $T_c$ has the companion exterior $E_K$ as its outside component. No conclusion is made for an essential companion torus that is absorbed into a Seifert-fibered union.

Ryan Budney, JSJ-decompositions of knot and link complements in the 3-sphere

Allen Hatcher, Notes on Basic 3-Manifold Topology, Theorem 1.9