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 K can be picturesquely described as follows: start by taking a nontrivial knot K' lying inside an unknotted solid torus V. Here "nontrivial" means that the knot K' is not allowed to sit inside of a 3-ball in V and 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 and. The central core curve of the solid torus V is sent to a knot H, which is called the "companion knot" and is thought of as the planet around which the "satellite knot" K orbits. The construction ensures that is a non-boundary parallel incompressible torus in the complement of 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 V is an unknotted solid torus, is a tubular neighbourhood of an unknot J. The 2-component link K' \cup J together with the embedding f is called the pattern associated to the satellite operation. A convention: people usually demand that the embedding is untwisted in the sense that f must send the standard longitude of V to the standard longitude of f(V). Said another way, given any two disjoint curves, f preserves their linking numbers i.e.:.

Basic families

When is a torus knot, then K is called a cable knot. Examples 3 and 4 are cable knots. The cable constructed with given winding numbers (m,n) from another knot K, is often called the (m,n) cable of K. If K' is a non-trivial knot in S^3 and if a compressing disc for V intersects K' in precisely one point, then K is called a connect-sum. Another way to say this is that the pattern K' \cup J is the connect-sum of a non-trivial knot K' with a Hopf link. If the link K' \cup J is the Whitehead link, K is called a Whitehead double. If f is untwisted, K is called an untwisted Whitehead double.

Examples

Examples 5 and 6 are variants on the same construction. They both have two non-parallel, non-boundary-parallel incompressible tori in their complements, splitting the complement into the union of three manifolds. In 5, those manifolds are: the Borromean rings complement, trefoil complement, and figure-8 complement. In 6, the figure-8 complement is replaced by another trefoil complement.

Origins

In 1949 Horst Schubert proved that every oriented knot in S^3 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 S^3 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.