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

<math>\int_C \kappa \,ds \leq 4\pi</math>

then C is an unknot, where <math>\kappa</math> 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 C in R³, the total curvature satisfies

<math>\int_C \kappa\,ds > 4\pi.</math>

The work of Fary and Milnor was independent. Legend has it that John Milnor was asleep in his math class at Princeton University when the professor wrote three unsolved knot theory problems on the board, one of which was the Fary-Milnor Theorem. Milnor, who was still an undergraduate, woke up at the end of class and wrote them down thinking they were assigned as homework. The next week, he turned in solutions for each of the problems, including a proof of this theorem.

[edit] References

• I. Fary, Sur la Coubure Totale d’une Courbe Gauche Faisant un Noeud. Bull. Soc. Math. France 77(1949) pp. 128-138.[
• J.W. Milnor, On the Total Curvature of Knots. Ann. of Math. 52 (1949), no. 2, pp. 248-257.