Satellite knot explained

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]

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^[3] 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,^[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
• Bing double

References

1. Colin Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, (2001),
2. Book: Handbook of Knot Theory . Menasco, William . William Menasco . Thistlethwaite, Morwen . Morwen Thistlethwaite . Elsevier . 2005 . 0080459544 . 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, http://www-bcf.usc.edu/~fbonahon/Research/Preprints/BonSieb.pdf
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