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In the [[knot theory|mathematical theory of knots]], a '''satellite knot''' is a [[knot (mathematics)|knot]] that contains an [[incompressible surface|incompressible]], non-[[boundary parallel]] [[torus]] in its [[knot complement|complement]].<ref>Colin Adams, ''The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots'',  (2001), ISBN 0-7167-4219-5</ref> Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include [[prime knot|composite]] knots, '''cable knots''' and '''Whitehead doubles.''' (''See'' [[#Basic families|Basic families]], below for definitions of the last two classes.)
[[Image:Sum of knots3.svg|thumb|right|Example 1: A connect-sum of a trefoil and figure-8 knot.]]

A satellite knot <math>K</math> can be picturesquely described as follows: start by taking a nontrivial knot <math>K'</math> lying inside an unknotted solid torus <math>V</math>.  Here "nontrivial" means that the knot <math>K'</math> is not allowed to sit inside of a 3-ball in <math>V</math> and  <math>K'</math> 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. [[Image:B sat2.png|thumb|right|Example 2: The Whitehead double of the figure-8.]]
This means there is a non-trivial embedding <math>f:V \to S^3</math> and <math>K=f(K')</math>.  The central core curve of the solid torus <math>V</math> is sent to a knot <math>H</math>, which is called the "companion knot" and is thought of as the planet around which the "satellite knot" <math>K</math> orbits.The construction ensures that <math>f(\partial V)</math> is a non-boundary parallel incompressible torus in the complement of <math>K</math>. 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.  [[Image:B sat3.png|thumb|right|Example 3: A cable of a connect-sum.]]

Since <math>V</math> is an unknotted solid torus, <math>S^3 \setminus V</math> is a tubular neighbourhood of an unknot <math>J</math>.  The 2-component link <math>K' \cup J</math> together with the embedding <math>f</math> is called the '''pattern''' associated to the satellite operation.  

A convention: people usually demand that the embedding <math>f : V \to S^3</math> is '''untwisted''' in the sense that <math>f</math> must send the standard longitude of <math>V</math> to the standard longitude of <math>f(V)</math>.  Said another way, given two disjoint curves <math>c_1,c_2 \subset V</math>, <math>f</math> must preserve their linking numbers i.e.:  <math>lk(f(c_1),f(c_2))=lk(c_1,c_2)</math>.

==Basic families==

When <math>K' \subset \partial V</math> is a [[torus knot]], then <math>K</math> is called a ''cable knot.'' Examples 3 and 4 are cable knots.

If <math>K'</math> is a non-trivial knot in <math>S^3</math> and if a compressing disc for <math>V</math> intersects <math>K'</math> in precisely one point, then <math>K</math> is called a ''connect-sum.''  Another way to say this is that the pattern <math>K' \cup J</math> is the connect-sum of a non-trivial knot <math>K'</math> with a Hopf link.

If the link <math>K' \cup J</math> is the [[Whitehead link]], <math>K</math> is called a ''Whitehead double.''  If <math>f</math> is untwisted, <math>K</math> is called an untwisted Whitehead double.

==Examples==

Example 1: The connect-sum of a figure-8 knot and trefoil. 

Example 2: Untwisted Whitehead double of a figure-8.

Example 3: Cable of a connect-sum. 

Example 4: Cable of trefoil.

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 Example 5 those manifolds are: the [[Borromean rings]] complement, trefoil complement and figure-8 complement. In Example 6 the figure-8 complement is replaced by another trefoil complement. 

[[Image:B sat4.png|thumb|right|Example 4: A cable of a trefoil.]]
[[Image:B sat1.png|thumb|right|Example 5: A knot which is a 2-fold satellite i.e.: it has non-parallel swallow-follow tori.]]
[[Image:Knot with borromean rings in jsj decomp.png|thumb|right|Example 6: A knot which is a 2-fold satellite i.e.: it has non-parallel swallow-follow tori.]]

==Origins==

In 1949 <ref>Schubert, H. Die eindeutige Zerlegbarkeit eines Knotens in Primknoten. S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), 57&ndash;104.</ref> [[Horst Schubert]] proved that every oriented knot in <math>S^3</math> 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 <math>S^3</math> 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'',<ref>Schubert, H. Knoten und Vollringe. Acta Math. 90 (1953), 131&ndash;286.</ref> 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.<ref>Waldhausen, F. On irreducible 3-manifolds which are sufficiently large.Ann. of Math. (2) 87 (1968), 56&ndash;88.</ref>  Waldhausen conjectured what is now the [[JSJ decomposition|Jaco&ndash;Shalen&ndash;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 conjecture|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.<ref>F.Bonahon, L.Siebenmann, New Geometric Splittings of Classical Knots, and the Classification and Symmetries of Arborescent Knots, [http://www-bcf.usc.edu/~fbonahon/Research/Preprints/BonSieb.pdf]</ref>

==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.<ref>Motegi, K. Knot Types of Satellite Knots and Twisted Knots. Lectures at Knots '96. World Scientific.</ref> With a suitably enhanced notion of satellite operation called splicing, the [[JSJ decomposition]] gives a proper uniqueness theorem for satellite knots.<ref>Eisenbud, D. Neumann, W. Three-dimensional link theory and invariants of plane curve singularities. Ann. of Math. Stud. 110</ref><ref>Budney, R. JSJ-decompositions of knot and link complements in S^3. L'enseignement Mathematique 2e Serie Tome 52 Fasc. 3&ndash;4 (2006). arXiv:math.GT/0506523</ref>

==See also==
*[[Hyperbolic knot]]
*[[Torus knot]]

==References==
<references/>

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[[Category:Satellite knots and links| ]]