In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 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 3-manifold; the boundary of ''X
K'' and the boundary of the neighborhood ''N'' are homeomorphic to a two-
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
. Sometimes the ambient manifold ''M'' is understood to be the
3-sphere. Context is needed to determine the usage. There are analogous definitions for the
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 complements then there is a homeomorphism of the three-sphere taking one knot to the other.
Knot complements are
Haken manifolds.
More generally complements of
links are Haken manifolds.
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.
References
Knot theory
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