# Knot 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*,

The knot complement *X _{K}* is a compact 3-manifold; the boundary of

*X*and the boundary of the neighborhood

_{K}*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.

## See also

## Further reading

- C. Gordon and J. Luecke, "Knots are determined by their Complements",
*J. Amer. Math. Soc.*,**2**(1989), 371–415.