In mathematics, the knot complement of a tame knot K is the threedimensional space surrounding the knot. To make this precise, suppose that K is a knot in a threemanifold M (most often, M is the 3sphere). 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 3manifold; the boundary of X_{K} and the boundary of the neighborhood N are homeomorphic to a twotorus. Sometimes the ambient manifold M is understood to be 3sphere. 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 threesphere 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 threesphere taking one knot to the other.
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Triangulating the figure 8 knot complement

Figure 8 knot complement

Not Knot
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See also
Further reading
 C. Gordon and J. Luecke, "Knots are determined by their Complements", J. Amer. Math. Soc., 2 (1989), 371–415.