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  "body": "\n## Definition\n\nLet $V = S^1 \\times D^2$ be the solid torus, and let $P \\subset \\operatorname{int} V$ be a tame [[knot|QYEAZTYGPT]], the *pattern*, such that:\n\n1. $P$ is not contained in any $3$-ball in $V$;\n2. $P$ is not isotopic in $V$ to the core circle $S^1 \\times \\set{0}$.\n\nLet $C \\subset S^3$ be a knot that is not the [[unknot|1TQ06R4Q66]], the *companion*.\nLet $h \\colon V \\to S^3$ be an embedding onto a closed tubular neighbourhood of $C$.\nA knot in $S^3$ is a *satellite knot* if it is isotopic to $h(P)$ for some such $P$, $C$ and $h$.\n\n## Remarks\n\nEquivalently, a knot $K$ is a satellite knot if $S^3 \\setminus K$ contains an incompressible torus that is not parallel to the boundary of a neighbourhood of $K$.\nExample: let $C$ be a [[trefoil knot|VYTYSVHN4A]] and let $P$ be the $(2, 1)$-curve on a smaller concentric torus in $V$.\nThen $h(P)$ is a cable of the trefoil and a satellite knot.\nNon-example: if $P$ is the core circle, clause 2 fails and $h(P)$ is just $C$.\nEvery [[knot sum|AGZSZW7GYE]] $K_1 \\# K_2$ of nontrivial knots is a satellite with companion $K_1$.\nWhitehead doubles are satellites whose pattern has winding number $0$ in $V$.\n",
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