Satellite knot

In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement.^[1] Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots, and Whitehead doubles. A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion.^[2]: 217

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.

This means there is a non-trivial embedding ${\displaystyle f\colon V\to S^{3}}$ and ${\displaystyle K=f\left(K'\right)}$. The central core curve of the solid torus ${\displaystyle V}$ is sent to a knot ${\displaystyle H}$, which is called the "companion knot" and is thought of as the planet around which the "satellite knot" ${\displaystyle K}$ orbits. The construction ensures that ${\displaystyle f(\partial V)}$ is a non-boundary parallel incompressible torus in the complement of ${\displaystyle K}$. Composite knots contain a certain kind of incompressible torus called a swallow-follow torus, which can be visualized as swallowing one summand and following another summand.

Since ${\displaystyle V}$ is an unknotted solid torus, ${\displaystyle S^{3}\setminus V}$ is a tubular neighbourhood of an unknot ${\displaystyle J}$. The 2-component link ${\displaystyle K'\cup J}$ together with the embedding ${\displaystyle f}$ is called the pattern associated to the satellite operation.

A convention: people usually demand that the embedding ${\displaystyle f\colon V\to S^{3}}$ is untwisted in the sense that ${\displaystyle f}$ must send the standard longitude of ${\displaystyle V}$ to the standard longitude of ${\displaystyle f(V)}$. Said another way, given any two disjoint curves ${\displaystyle c_{1},c_{2}\subset V}$, ${\displaystyle f}$ preserves their linking numbers i.e.: ${\displaystyle \operatorname {lk} (f(c_{1}),f(c_{2}))=\operatorname {lk} (c_{1},c_{2})}$.