The Fary-Milnor theorem states that the total curvature of a knotted simple closed curve in R^3 is greater than 4π. That is, let γ:[0,l]→R^3 be isotopic to S^1 and be parametrized by arc length s with curvature k(s), then ∫|k(s)|ds>4π.e are going to show this theorem for simple closed ploygons since a simple closed curve of finite total curvature is isotopic to an inscribed polygon, and the total curvature of an inscribed polygon never exceeds that of the original curve.
