A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion.
can be picturesquely described as follows: start by taking a nontrivial knot
lying inside an unknotted solid torus
Here "nontrivial" means that the knot
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.
The central core curve of the solid torus
is a non-boundary parallel incompressible torus in the complement of
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.
is a tubular neighbourhood of an unknot
is called the pattern associated to the satellite operation.
A convention: people usually demand that the embedding
must send the standard longitude of
preserves their linking numbers i.e.:
is called a cable knot.
is called an untwisted Whitehead double.
They both have two non-parallel, non-boundary-parallel incompressible tori in their complements, splitting the complement into the union of three manifolds.
In 1949[3] Horst Schubert proved that every oriented knot in
decomposes as a connect-sum of prime knots in a unique way, up to reordering, making the monoid of oriented isotopy-classes of knots in
a free commutative monoid on countably-infinite many generators.
Shortly after, he realized he could give a new proof of his theorem by a close analysis of the incompressible tori present in the complement of a connect-sum.
This led him to study general incompressible tori in knot complements in his epic work Knoten und Vollringe,[4] where he defined satellite and companion knots.
Schubert's demonstration that incompressible tori play a major role in knot theory was one several early insights leading to the unification of 3-manifold theory and knot theory.
It attracted Waldhausen's attention, who later used incompressible surfaces to show that a large class of 3-manifolds are homeomorphic if and only if their fundamental groups are isomorphic.
[5] Waldhausen conjectured what is now the Jaco–Shalen–Johannson-decomposition of 3-manifolds, which is a decomposition of 3-manifolds along spheres and incompressible tori.
This later became a major ingredient in the development of geometrization, which can be seen as a partial-classification of 3-dimensional manifolds.
The ramifications for knot theory were first described in the long-unpublished manuscript of Bonahon and Siebenmann.
[6] In Knoten und Vollringe, Schubert proved that in some cases, there is essentially a unique way to express a knot as a satellite.
But there are also many known examples where the decomposition is not unique.
[7] With a suitably enhanced notion of satellite operation called splicing, the JSJ decomposition gives a proper uniqueness theorem for satellite knots.