In mathematics, the knot complement of a

tame knot Tame may refer to: *Taming, the act of training wild animals * River Tame, Greater Manchester *River Tame, West Midlands and the Tame Valley * Tame, Arauca, a Colombian town and municipality * "Tame" (song), a song by the Pixies from their 1989 a ...

''K'' is the space where the knot is not. If a knot is embedded in the

3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...

, 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

). 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 A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...

of ''N'', :

X_K = M - \mbox(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 British ...

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 ...

; 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 not tou ...

. Sometimes the ambient manifold ''M'' is understood to be

. Context is needed to determine the usage. There are analogous definitions of link complement. Many

knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...

s, such as the

knot group In mathematics, a knot (mathematics), knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3, :\pi_1(\mathbb^3 \setminus K). Oth ...

, 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 In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a ...

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.

See also

Further reading