Fary-Milnor theorem

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In mathematics, the Fary-Milnor theorem in knot theory states that for any knot K in R³, if the total curvature

then K is an unknot, where κ is the curvature (it is possible for an unknotted curve to have large total curvature). As corollary to the Fary-Milnor theorem, for any knotted curve K in R³, the total curvature satisfies

The theorem was proved independently by István Fáry in 1949 and John Milnor in 1950.

[edit] References

• I. Fary, Sur la courbure totale d’une courbe gauche faisant un nœud, Bulletin de la Société Mathématique de France 77 (1949), pp. 128–138.
• J.W. Milnor, On the total curvature of knots, Annals of Mathematics 52 (1950), no. 2, pp. 248–257.