Knot complement

Unknot complement is homeomorphic to a solid torus.
Hopf band is one Hopf link complement.

In mathematics, the knot complement of a tame knot K is the three-dimensional space surrounding 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,

${\displaystyle X_{K}=M-{\mbox{interior}}(N).}$

The knot complement XK is a compact 3-manifold; the boundary of XK and the boundary of the neighborhood N are homeomorphic to a two-torus. 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 complements then there is a homeomorphism of the three-sphere taking one knot to the other.