Satellite pattern, companion, and winding number

Let $V=S^1\times D^2$ be the standard oriented solid torus, with oriented core $S^1\times \{0\}$. A pattern is a smooth oriented knot $P$ contained in the interior of $V$, together with a chosen closed tubular neighbourhood $N(P)$ contained in the interior of $V$. Its winding number $w(P)$ is the integer by which the homology class $[P]$ maps to $H_1(V;\mathbb{Z})\cong \mathbb{Z}$ under the isomorphism that sends the oriented core to $1$. The pattern exterior is $X_P=V\setminus \mathrm{int}N(P)$; its outer torus is $\mathrm{\partial }V$ and its inner torus is $\mathrm{\partial }N(P)$. A companion is a smooth oriented knot $K\subset S^3$ with a specified orientation-preserving tubular embedding $e:V\to S^3$ that sends the core to $K$ and identifies the solid-torus meridian and product longitude with the positive meridian and preferred longitude of $K$. The satellite knot $P(K)$ is $e(P)$, with chosen exterior $E_{P(K)}=S^3\setminus \mathrm{int}e(N(P))$; the image $e(\mathrm{\partial }V)$ is its companion torus. The outer torus is incompressible when it bounds no properly embedded disk in $X_P$ with essential boundary on $\mathrm{\partial }V$. It is product-parallel to the inner torus when $X_P$ is a product $T^2\times [0,1]$ with those tori as its ends. This definition alone asserts neither incompressibility nor essentiality of the companion torus.

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

Allen Hatcher, Notes on Basic 3-Manifold Topology