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

\oint_K \kappa \,ds \leq 4\pi

then K is an unknot, where \kappa 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 curves K in R³, the total curvature satisfies

\oint_K \kappa\,ds > 4\pi.

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

References

• I. Fary, ''Sur la courbure totale d?une courbe gauche faisant un n?ud'', 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.

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