Math 2.0

Differential geometry Curves in the plane and in space Theorem

Fáry–Milnor theorem

Builds on Total curvatureCurvature of a curveKnotClosed curve

Statement

Let γ ⁣:[a,b]→ℝ3\gamma \colon [a, b] \to \R^3 be a regular simple closed curve of class C2C^2 whose derivatives up to order 22 also agree at aa and bb. If the total curvature of γ\gamma is at most 4π4\pi, then γ\gamma is unknotted, that is, its image is a trivial knot. Equivalently, every knotted curve of this kind has total curvature greater than 4π4\pi.

Remarks

Milnor's proof uses height functions. For almost every unit vector vv, the function t↦⟨γ(t),v⟩t \mapsto \ip{\gamma(t)}{v} has finitely many local maxima, and the total curvature equals 2π2\pi times the average number of these maxima over all vv. If the total curvature is less than 4π4\pi, some direction vv gives exactly one local maximum and one local minimum. Then each plane orthogonal to vv meets the curve in at most two points, and joining these pairs of points gives an embedded disc bounded by the curve. So the curve is unknotted; the case of total curvature exactly 4π4\pi needs a further argument. Example: a standard trefoil knot must have total curvature greater than 4π4\pi, while a round circle, with 2π2\pi, is unknotted. István Fáry and John Milnor proved the theorem independently, around 1949 and 1950.

Going deeper

For a student

The theorem refines Fenchel's theorem, which gives the bound 2π2\pi for every closed curve. The bound 4π4\pi is sharp: there are knotted curves with total curvature as close to 4π4\pi as one likes. Milnor's averaging formula for the total curvature is a model of later integral-geometric methods, such as the Crofton formula.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

Show the path as a graph
Curvature of a curve, concept, no evidence levelCurvature of a curveconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptKnot, concept, no evidence levelKnotconceptTotal curvature, concept, no evidence levelTotal curvatureconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionClosed curve, definition, no evidence levelClosed curvedefinitionFáry–Milnor theorem, statement, no evidence levelFáry–Milnor 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. Curvature of a curve concept
  2. Euclidean space object
  3. Function concept
  4. Knot concept
  5. Total curvature concept
  6. Fréchet derivative definition
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Limit of a function of several variables concept
  10. Continuity in R^n definition
  11. Differentiable function of several variables definition
  12. Partial derivative definition
  13. Differentiable on an open set definition
  14. Higher partial derivatives definition
  15. Iterated derivative definition
  16. C^k function characterised by continuous derivatives statement
  17. Parametrised curve definition
  18. Closed curve definition
  19. Fáry–Milnor theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/F%C3%A1ry%E2%80%93Milnor_theorem
About this page
TagB1Z1C8AHVD
URIm2://B1Z1C8AHVD
Kindstatement, theorem
WritingW0 stub
MSC53A04, 57K10
StatusThis copy of the site has no build data. Run m2 status B1Z1C8AHVD.
AboutTotal curvature, Knot
UsesTotal curvature, Curvature of a curve, Knot, Closed curve
Same as
Changed2 November 2026

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