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 $$K$$ 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 $$f\colon V \to S^3$$ and $$K = f\left(K'\right)$$. 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 $$f(\partial V)$$ 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, $$S^3 \setminus V$$ 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 $$f \colon V \to S^3$$ 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 $$c_1, c_2 \subset V$$, $$f$$ preserves their linking numbers i.e.: $$\operatorname{lk}(f(c_1), f(c_2)) = \operatorname{lk}(c_1, c_2)$$.

Basic families
When $$K' \subset \partial V$$ 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.