mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 knot Tame may refer to: *Taming, the act of training wild animals * River Tame, Greater Manchester *River Tame, West Midlands and the Tame Valley * Tame, Arauca, a Colombian town and municipality * "Tame" (song), a song by the Pixies from their 1989 a ...

s 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 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...

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 Marc Edward Culler (born November 22, 1953) is an American mathematician who works in geometric group theory and low-dimensional topology. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and ...

and

Peter Shalen Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition. Life He graduated from Stuyvesant High School in 1962, and went on to earn a B.A. from Harvard Coll ...

on the

cyclic surgery theorem In three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold ''M'' whose boundary is a torus ''T'', if ''M'' is not a Seifert-fibered space A S ...

, combinatorial techniques in the style of Litherland,

thin position Thin may refer to: * a lean body shape. ''(See also: emaciation, underweight)'' * ''Thin'' (film), a 2006 HBO documentary about eating disorders * Paper Thin (disambiguation), referring to multiple songs * Thin (web server), a Ruby web-server b ...

, and Scharlemann cycles. For link complements, it is not in fact true that links are determined by their complements. For example,

JHC Whitehead John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princeton ...

proved that there are infinitely many links whose complements are all homeomorphic to the

Whitehead link In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop. Structure A common w ...

. 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