“Satellite knot”的意思、由来-开放百科全书

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. (See Basic families, below for definitions of the last two classes.) 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]{{rp|217}}

A satellite knot

can be picturesquely described as follows: start by taking a nontrivial knot

lying inside an unknotted solid torus

. Here "nontrivial" means that the knot

is not allowed to sit inside of a 3-ball in

and

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

is sent to a knot

, which is called the "companion knot" and is thought of as the planet around which the "satellite knot"

orbits.The construction ensures that

is a non-boundary parallel incompressible torus in the complement of

. 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

is an unknotted solid torus,

is a tubular neighbourhood of an unknot

. The 2-component link

together with the embedding

is called the pattern associated to the satellite operation.

A convention: people usually demand that the embedding

is untwisted in the sense that

must send the standard longitude of

to the standard longitude of

. Said another way, given any two disjoint curves

preserves their linking numbers i.e.:

Basic families

When

is a torus knot, then

is called a cable knot. Examples 3 and 4 are cable knots.

is a non-trivial knot in

and if a compressing disc for

intersects

in precisely one point, then

is called a connect-sum. Another way to say this is that the pattern

is the connect-sum of a non-trivial knot

with a Hopf link.

If the link

is the Whitehead link,

is called a Whitehead double. If

is untwisted,

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 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.

Origins

In 1949 ^[3] Horst Schubert proved that every oriented knot in

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

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

References

1. ^Colin Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, (2001), {{isbn|0-7167-4219-5}}
2. ^{{Cite book |title=Handbook of Knot Theory |url=https://books.google.com/books?id=EyYWVnK5z44C&pg=PA217&lpg=PA217&dq=satellite+link+knots&source=bl&ots=wW7-HL_6Px&sig=mtZT25G9-MOzn27JftBn8A7WP-I&hl=en&sa=X&ei=4kbxU_4EpN_wAbfygegB&ved=0CE8Q6AEwCw#v=onepage&q=satellite%20link%20knots&f=false |editor1=Menasco, William |editorlink1=William Menasco |editor2=Thistlethwaite, Morwen |editorlink2=Morwen Thistlethwaite |publisher=Elsevier |year=2005 |isbn=0080459544 |accessdate=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