Total curvature of graphs in space

Abstract

The Fary-Milnor Theorem says that any embedding of the circle S1 into R3 of total curvature less than 4π is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topological type of graphs G{cyrillic}, what limitations on the isotopy class of G{cyrillic} are implied by a bound on total curvature? Especially: what does "total curvature" mean for a graph? I shall discuss several natural notions of the total curvature of a graph. Turning to the problem of isotopy type, I shall then focus on the notion of net total curvature N(G{cyrillic}) of a graph G{cyrillic} ⊂ R3, and outline the proof that if G{cyrillic} is homeomorphic to the θ-graph, then N(G{cyrillic}) ≥ 3 π; and if N(G{cyrillic}) < 4 π, then G{cyrillic} is isotopic in R3 to a planar θ-graph. Proofs will be given in full in [GY2].