In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the knot complement of a
tame knot ''K'' is the space where the knot is not. If a knot is embedded in the
3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a knot in a three-manifold ''M'' (most often, ''M'' is the
3-sphere). Let ''N'' be a
tubular neighborhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the ...
of ''K''; so ''N'' is a
solid torus. The knot complement is then the
complement of ''N'',
:
The knot complement ''X
K'' is a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
3-manifold; the boundary of ''X
K'' and the boundary of the neighborhood ''N'' are homeomorphic to a two-
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 ...
. Sometimes the ambient manifold ''M'' is understood to be
3-sphere. Context is needed to determine the usage. There are analogous definitions of
link complement.
Many
knot invariants, such as the
knot group, are really invariants of the complement of the knot. When the ambient space is the three-sphere no information is lost: the
Gordon–Luecke theorem states that a knot is determined by its complement. That is, if ''K'' and ''K''′ are two knots with
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
complements then there is a homeomorphism of the three-sphere taking one knot to the other.
See also
*
Knot genus
*
Seifert surface
Further reading
* C. Gordon and J. Luecke, "Knots are determined by their Complements", ''
J. Amer. Math. Soc.'', 2 (1989), 371–415.
Knot theory
{{knottheory-stub