Total curvature of graphs after Milnor and Euler (Q433551)

A theorem due to Milnor and Fáry (independently proven in 1949) states that a curve in \(\mathbb R^3\) whose total curvature is at most \(4\pi\) is unknotted. In the paper under review, the authors extend Milnor's methods to include the case of a finite graph embedded in \(\mathbb{R}^n\). They introduce a new notion of total curvature which they call net total curvature and investigate its properties. The definition is first given for piecewise \(C^2\) graphs as the total curvature of the smooth arcs, and the contribution at the vertices is taken into account. The definition is then extended to continuously embedded graphs. The methods used in the paper include a Crofton-type formula, as in Milnor's work, and the consideration of the double cover of a graph as an Eulerian circuit. The authors obtain results on upper/lower bounds of the total curvature on isotopy/homotopy classes of embeddings, which show that the new notion of net total curvature gives an efficient measure of the complexity of spatial graphs in differential-geometric terms.