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 thickened neighborhood of K; so N is a solid torus. The knot complement is then the complement of N,

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

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. 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.

References

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