Math 2.0

Manifolds and cell complexes Knot theory Problem

Unknotting problem

Builds on KnotUnknotReidemeister's theorem

Problem

The input is a finite diagram of a tame knot in ℝ3\R^3, for example a polygon with finitely many vertices given by rational coordinates. The question is whether the knot is the unknot, that is, whether it is ambient isotopic to a round circle. The problem asks for an algorithm that answers this question for every input, and for the cost of such an algorithm.

Remarks

Dehn's work of 1910 began the search for such an algorithm, through the fundamental group of the knot complement. Haken proved with his theory of normal surfaces, published in 1961, that the problem is decidable. His method searches for a disk bounded by the knot among finitely many normal surfaces in the knot complement. Hass, Lagarias and Pippenger proved in 1999 that the problem lies in the complexity class NP. Lackenby proved without extra hypotheses that it also lies in co-NP. It is not known whether the problem can be solved in polynomial time. Reidemeister moves give another approach: a diagram of the unknot can be reduced to a circle by such moves, and there are explicit bounds on the number of moves needed.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Knot, concept, no evidence levelKnotconceptLink, concept, no evidence levelLinkconceptUnknot, example, no evidence levelUnknotexampleReidemeister's theorem, statement, no evidence levelReidemeister's theoremstatementUnknotting problem, problem, no evidence levelUnknotting problemproblem
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. Knot concept
  2. Link concept
  3. Unknot example
  4. Reidemeister's theorem statement
  5. Unknotting problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Unknotting_problem
  2. Web https://arxiv.org/abs/math/9807016, Section 1
About this page
TagVXSD46NRX0
URIm2://VXSD46NRX0
Kindproblem
WritingW0 stub
MSC57K10, 68Q17
StatusThis copy of the site has no build data. Run m2 status VXSD46NRX0.
AboutKnot
UsesKnot, Unknot, Reidemeister's theorem
Same as
Changed2 November 2026

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