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. 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. A satellite knot formula_1 can be picturesquely described as follows: start by taking a nontrivial knot formula_2 lying inside an unknotted solid torus formula_3. Here "nontrivial" means  that the knot formula_2 is not allowed to sit inside of a 3-ball in formula_3 and formula_2 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 formula_7 and formula_8. The central core curve of the solid torus formula_3 is sent to a knot formula_10, which is called the "companion knot" and is thought of as the planet around which the "satellite knot" formula_1 orbits. The construction ensures that formula_12 is a non-boundary parallel incompressible  torus in the complement of formula_1. 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 formula_3 is an unknotted solid torus, formula_15 is a tubular neighbourhood of an unknot formula_16. The 2-component link formula_17 together with the embedding formula_18 is called the "pattern" associated to the satellite operation. A convention: people usually demand that the embedding formula_19 is "untwisted" in the sense that formula_18 must send the standard longitude of formula_3 to the standard longitude of formula_22. Said another way, given  any two disjoint curves formula_23, formula_18 preserves their linking numbers i.e.: formula_25. Basic families. When formula_26 is a torus knot, then formula_1 is called a "cable knot". Examples 3 and 4 are cable knots. If formula_2 is a non-trivial knot in formula_29 and if a compressing disc for formula_3 intersects formula_2 in precisely one point, then formula_1 is called a "connect-sum". Another way to say this is that the pattern formula_17 is the connect-sum of a non-trivial knot formula_2 with a Hopf link. If the link formula_17 is the Whitehead link, formula_1 is called a "Whitehead  double". If formula_18 is untwisted, formula_1 is called an untwisted Whitehead double. Origins. In 1949 Horst Schubert proved that every oriented knot in formula_29 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 formula_29 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", where he defined satellite and companion knots. Follow-up work. 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. 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. Uniqueness of satellite decomposition. 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. With a suitably enhanced notion of satellite operation called splicing, the JSJ decomposition gives a proper uniqueness theorem for satellite  knots.