The total curvature of a smooth simple closed curve in Euclidean 3-space  R3  is always  ≥  2 pi ,  with equality only for plane convex curves.

By contrast, the Fary-Milnor theorem states that if the curve is knotted, then its total curvature must be more than double this, thus > 4 pi .

We will outline a proof of this result, and then make a few remarks about its generalization to knotted surfaces in Euclidean 4-space R4 .