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. ^[1] 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. ^{[2] : 217}

A satellite knot ${\displaystyle K}$ can be picturesquely described as follows: start by taking a nontrivial knot ${\displaystyle K'}$ lying inside an unknotted solid torus ${\displaystyle V}$. Here "nontrivial" means that the knot ${\displaystyle K'}$ is not allowed to sit inside of a 3-ball in ${\displaystyle V}$ and ${\displaystyle K'}$ 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 ${\displaystyle f\colon V\to S^{3}}$ and ${\displaystyle K=f\left(K'\right)}$. The central core curve of the solid torus ${\displaystyle V}$ is sent to a knot ${\displaystyle H}$, which is called the "companion knot" and is thought of as the planet around which the "satellite knot" ${\displaystyle K}$ orbits. The construction ensures that ${\displaystyle f(\partial V)}$ is a non-boundary parallel incompressible torus in the complement of ${\displaystyle K}$. 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 ${\displaystyle V}$ is an unknotted solid torus, ${\displaystyle S^{3}\setminus V}$ is a tubular neighbourhood of an unknot ${\displaystyle J}$. The 2-component link ${\displaystyle K'\cup J}$ together with the embedding ${\displaystyle f}$ is called the pattern associated to the satellite operation.

A convention: people usually demand that the embedding ${\displaystyle f\colon V\to S^{3}}$ is untwisted in the sense that ${\displaystyle f}$ must send the standard longitude of ${\displaystyle V}$ to the standard longitude of ${\displaystyle f(V)}$. Said another way, given any two disjoint curves ${\displaystyle c_{1},c_{2}\subset V}$, ${\displaystyle f}$ preserves their linking numbers i.e.: ${\displaystyle \operatorname {lk} (f(c_{1}),f(c_{2}))=\operatorname {lk} (c_{1},c_{2})}$.

Basic families

When ${\displaystyle K'\subset \partial V}$ is a torus knot, then ${\displaystyle K}$ 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 ${\displaystyle K'}$ is a non-trivial knot in ${\displaystyle S^{3}}$ and if a compressing disc for ${\displaystyle V}$ intersects ${\displaystyle K'}$ in precisely one point, then ${\displaystyle K}$ is called a connect-sum. Another way to say this is that the pattern ${\displaystyle K'\cup J}$ is the connect-sum of a non-trivial knot ${\displaystyle K'}$ with a Hopf link.

If the link ${\displaystyle K'\cup J}$ is the Whitehead link, ${\displaystyle K}$ is called a Whitehead double. If ${\displaystyle f}$ is untwisted, ${\displaystyle K}$ is called an untwisted Whitehead double.

Examples

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  Example 1: A connect-sum of a trefoil and figure-8 knot.
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  Example 2: The Whitehead double of the figure-8.
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  Example 3: A cable of a connect-sum.
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  Example 4: A cable of a trefoil.
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  Example 5: A knot which is a 2-fold satellite i.e.: it has non-parallel swallow-follow tori.
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  Example 6: A knot which is a 2-fold satellite i.e.: it has non-parallel swallow-follow tori.

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 ^[3] Horst Schubert proved that every oriented knot in ${\displaystyle S^{3}}$ 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 ${\displaystyle S^{3}}$ 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, ^[4] 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. ^[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]

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. ^[7] With a suitably enhanced notion of satellite operation called splicing, the JSJ decomposition gives a proper uniqueness theorem for satellite knots. ^{[8] [9]}

See also

• Hyperbolic knot
• Torus knot

References

1. Colin Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, (2001), ISBN   0-7167-4219-5
2. Menasco, William; Thistlethwaite, Morwen, eds. (2005). Handbook of Knot Theory. Elsevier. ISBN   0080459544 . Retrieved 2014-08-18.
3. Schubert, H. Die eindeutige Zerlegbarkeit eines Knotens in Primknoten. S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), 57–104.
4. Schubert, H. Knoten und Vollringe. Acta Math. 90 (1953), 131–286.
5. Waldhausen, F. On irreducible 3-manifolds which are sufficiently large.Ann. of Math. (2) 87 (1968), 56–88.
6. F.Bonahon, L.Siebenmann, New Geometric Splittings of Classical Knots, and the Classification and Symmetries of Arborescent Knots,
7. Motegi, K. Knot Types of Satellite Knots and Twisted Knots. Lectures at Knots '96. World Scientific.
8. Eisenbud, D. Neumann, W. Three-dimensional link theory and invariants of plane curve singularities. Ann. of Math. Stud. 110
9. Budney, R. JSJ-decompositions of knot and link complements in S^3. L'enseignement Mathematique 2e Serie Tome 52 Fasc. 3–4 (2006). arXiv:math.GT/0506523