The Second Hull of a Knotted Curve
arXiv:math/0204106 · doi:10.1353/ajm.2003.0038
Abstract
The convex hull of a set K in space consists of points which are, in a certain sense, "surrounded" by K. When K is a closed curve, we define its higher hulls, consisting of points which are "multiply surrounded" by the curve. Our main theorem shows that if a curve is knotted then it has a nonempty second hull. This provides a new proof of the Fary/Milnor theorem that every knotted curve has total curvature at least 4pi.
7 pages, 6 figures; final version (only minor changes) to appear in Amer.J.Math