Fary-Milnor theorem

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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

\int_C \kappa \,ds \leq 4\pi

then C 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 C in R3, the total curvature satisfies

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

The work of Fary and Milnor was independent.

[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.
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