On the total curvature of semialgebraic graphs

Abstract

We define the total curvature of a semialgebraic embedding of a graph in the 3-dimensional Euclidean space. We prove that it satisfies a Chern-Lashof type inequality and we describe when the equality holds. We also prove a generalization of a classical result of Fary and Milnor stating that certain graphs cannot be knotted if they are not too curved. The techniques employed are Morse theoretic.