Whitehead double

In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots and Whitehead doubles. (See Basic families, below for definitions of the last two classes.) A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion.

A satellite knot ${\displaystyle K}$ can be picturesquely described as follows: start by taking a nontrivial knot ${\displaystyle K'}$ lying inside an unknotted solid torus ${\displaystyle V}$. Here "nontrivial" means that the knot ${\displaystyle K'}$ is not allowed to sit inside of a 3-ball in ${\displaystyle V}$ and ${\displaystyle K'}$ is not allowed to be isotopic to the central core curve of the solid torus. Then tie up the solid torus into a nontrivial knot.

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