Fary-Milnor theorem

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

:$oint\_K\; kappa\; ,ds\; leq\; 4pi$

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

:$oint\_K\; kappa,ds\; >\; 4pi.$

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

References

* I. Fary, [http://www.numdam.org/item?id=BSMF_1949__77__128_0 "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, [http://links.jstor.org/sici?sici=0003-486X%28195009%292%3A52%3A2%3C248%3AOTTCOK%3E2.0.CO%3B2-R "On the total curvature of knots"] , Annals of Mathematics 52 (1950), no. 2, pp. 248–257.