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
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
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 Whitehead double.
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.