In the
mathematical theory of knots, a satellite knot is a
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ...
that contains an
incompressible, non
boundary-parallel torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
in its
complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials ...
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
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. The cable constructed with given winding numbers (m,n) from another knot K, is often called ''the'' (m,n) cable of K.
If
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
In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop.
Structure
A common way ...
,
is called a ''Whitehead double''. If
is untwisted,
is called an untwisted Whitehead double.
Examples
# The connect-sum of a figure-8 knot and trefoil.
# Untwisted Whitehead double of a figure-8.
# Cable of a connect-sum.
# Cable of trefoil.
: 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
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'', 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
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
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 In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
: Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isot ...
gives a proper uniqueness theorem for satellite knots.
[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]
See also
*
Hyperbolic knot
*
Torus knot
References
{{Knot theory, state=collapsed