mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Gordon–Luecke theorem on

knot complement In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...

s states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian. The theorem is usually stated as "knots are determined by their complements"; however this is slightly ambiguous as it considers two knots to be equivalent if there is a self-homeomorphism taking one knot to the other. Thus mirror images are neglected. Often two knots are considered equivalent if they are '' isotopic''. The correct version in this case is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic. These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial

Dehn surgery In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...

on a nontrivial knot in the

3-sphere In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...

can yield the

. The theorem was proved by Cameron Gordon and John Luecke. Essential ingredients of the proof are their joint work with Marc Culler and Peter Shalen on the cyclic surgery theorem, combinatorial techniques in the style of Litherland, thin position, and Scharlemann cycles. For link complements, it is not in fact true that links are determined by their complements. For example, JHC Whitehead proved that there are infinitely many links whose complements are all homeomorphic to the Whitehead link. His construction is to twist along a disc spanning an unknotted component (as is the case for either component of the Whitehead link). Another method is to twist along an annulus spanning two components. Gordon proved that for the class of links where these two constructions are not possible there are finitely many links ''in this class'' with a given complement.

References

*Cameron Gordon and John Luecke, ''Knots are determined by their complements''. J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. *Cameron Gordon, ''Links and their complements.'' Topology and geometry: commemorating SISTAG, 71–82, Contemp. Math., 314, Amer. Math. Soc., Providence, RI, 2002. {{DEFAULTSORT:Gordon-Luecke theorem Knot theory 3-manifolds Theorems in topology