Curves, Knots, and Total Curvature

Charles Evans We present an exposition of various results dealing with the total curvature of curves in Euclidean 3-space. There are two primary results: Fenchel’s theorem and the theorem of Fary and Milnor. Fenchel’s theorem states that the total curvature of a simple closed curve is greater than or equal to 2π, with equality if and only if the curve is planar convex. The Fary-Milnor theorem states that the total curvature of a simple closed knotted curve is strictly greater than 4π. Several methods of proof are supplied, utilizing both curve-theoretic and surface-theoretic techniques, surveying methods from both differential and integral geometry. Related results are considered: the connection between total curvature and bridge number; an analysis of total curvature plus total torsion; a lower bound on the length of the normal indicatrix.