Knot complement

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In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. This solid torus is a thickened neighborhood of K. Note that the knot complement is a compact 3-manifold with boundary homeomorphic to a torus. Sometimes "knot complement" means the complement in the 3-sphere of a knot (whether tame or not), in which case the knot complement is not compact. 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. This is not a disadvantage because a theorem of Cameron Gordon and John Luecke 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 3-sphere taking one knot to the other. The proof requires many deep ideas -- among these is the notion of thin position due to David Gabai.

[edit] Reference

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