Math 2.0

Manifolds and cell complexes Knot theory Theorem

Gordon–Luecke theorem

Builds on KnotHomeomorphism

Statement

Let KK and K′K' be tame knots in S3S^3. Suppose that there is a homeomorphism S3∖K→S3∖K′S^3 \setminus K \to S^3 \setminus K'. Then there is a homeomorphism h ⁣:S3→S3h \colon S^3 \to S^3 with h(K)=K′h(K) = K'.

Remarks

The proof reduces the theorem to a statement about Dehn surgery. A homeomorphism of complements that does not carry meridian to meridian would give a nontrivial surgery on K′K' that yields S3S^3. Gordon and Luecke showed that a nontrivial Dehn surgery on a nontrivial knot never gives S3S^3, using a careful study of the graphs formed by intersecting planar surfaces. The homeomorphism hh may reverse orientation, so a knot and its mirror image have homeomorphic complements and count as equivalent here. The analogue for links fails: twisting a link along an unknotted component changes the link but not the homeomorphism type of its complement. The knot group alone does not determine the knot: the square knot and the granny knot have isomorphic groups but are different knots.

Going deeper

For a student

The theorem justifies studying a knot through its complement, for instance through the hyperbolic structure of the complement of a hyperbolic knot. Combined with Mostow–Prasad rigidity, it shows that a hyperbolic knot is determined by the isometry type of its complement. It is close to the Property P question: both are about which surgeries on a knot can give simple manifolds. For links, and for knots in other 33-manifolds, the complement can lose information.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Homeomorphism, concept, no evidence levelHomeomorphismconceptKnot, concept, no evidence levelKnotconceptGordon–Luecke theorem, statement, no evidence levelGordon–Luecke theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Homeomorphism concept
  2. Knot concept
  3. Gordon–Luecke theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Gordon%E2%80%93Luecke_theorem
About this page
TagEBASBW5TYS
URIm2://EBASBW5TYS
Kindstatement, theorem
WritingW0 stub
MSC57K10, 57K30
StatusThis copy of the site has no build data. Run m2 status EBASBW5TYS.
AboutKnot
UsesKnot, Homeomorphism
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Changed2 November 2026

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