- Source: 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. 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.: 217
A satellite knot
K
{\displaystyle K}
can be picturesquely described as follows: start by taking a nontrivial knot
K
′
{\displaystyle K'}
lying inside an unknotted solid torus
V
{\displaystyle V}
. Here "nontrivial" means that the knot
K
′
{\displaystyle K'}
is not allowed to sit inside of a 3-ball in
V
{\displaystyle V}
and
K
′
{\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
f
:
V
→
S
3
{\displaystyle f\colon V\to S^{3}}
and
K
=
f
(
K
′
)
{\displaystyle K=f\left(K'\right)}
. The central core curve of the solid torus
V
{\displaystyle V}
is sent to a knot
H
{\displaystyle H}
, which is called the "companion knot" and is thought of as the planet around which the "satellite knot"
K
{\displaystyle K}
orbits. The construction ensures that
f
(
∂
V
)
{\displaystyle f(\partial V)}
is a non-boundary parallel incompressible torus in the complement of
K
{\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
V
{\displaystyle V}
is an unknotted solid torus,
S
3
∖
V
{\displaystyle S^{3}\setminus V}
is a tubular neighbourhood of an unknot
J
{\displaystyle J}
. The 2-component link
K
′
∪
J
{\displaystyle K'\cup J}
together with the embedding
f
{\displaystyle f}
is called the pattern associated to the satellite operation.
A convention: people usually demand that the embedding
f
:
V
→
S
3
{\displaystyle f\colon V\to S^{3}}
is untwisted in the sense that
f
{\displaystyle f}
must send the standard longitude of
V
{\displaystyle V}
to the standard longitude of
f
(
V
)
{\displaystyle f(V)}
. Said another way, given any two disjoint curves
c
1
,
c
2
⊂
V
{\displaystyle c_{1},c_{2}\subset V}
,
f
{\displaystyle f}
preserves their linking numbers i.e.:
lk
(
f
(
c
1
)
,
f
(
c
2
)
)
=
lk
(
c
1
,
c
2
)
{\displaystyle \operatorname {lk} (f(c_{1}),f(c_{2}))=\operatorname {lk} (c_{1},c_{2})}
.
Basic families
When
K
′
⊂
∂
V
{\displaystyle K'\subset \partial V}
is a torus knot, then
K
{\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
K
′
{\displaystyle K'}
is a non-trivial knot in
S
3
{\displaystyle S^{3}}
and if a compressing disc for
V
{\displaystyle V}
intersects
K
′
{\displaystyle K'}
in precisely one point, then
K
{\displaystyle K}
is called a connect-sum. Another way to say this is that the pattern
K
′
∪
J
{\displaystyle K'\cup J}
is the connect-sum of a non-trivial knot
K
′
{\displaystyle K'}
with a Hopf link.
If the link
K
′
∪
J
{\displaystyle K'\cup J}
is the Whitehead link,
K
{\displaystyle K}
is called a Whitehead double. If
f
{\displaystyle f}
is untwisted,
K
{\displaystyle 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 Horst Schubert proved that every oriented knot in
S
3
{\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
S
3
{\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, 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. 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 gives a proper uniqueness theorem for satellite knots.
See also
Hyperbolic knot
Torus knot
Bing double
References
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- Liv Tyler
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- Satellite knot
- Hyperbolic link
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- Knot (mathematics)
- Knot theory
- List of knot theory topics
- Conway knot
- Trefoil knot
- JSJ decomposition
- Knot complement
