2008/06/02 by Robert Gulliver, Gulliver, Robert, Sumio Yamada +1 · 1 citation
Mathematics · #53A04 #58K99 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #math.GT #msc:53A04 #msc:58K99
paper · pdf · doi:10.48550/arxiv.0806.0406
22 pages, 2 figures in .eps format
arxiv created 2008/06/02 · openalex publication_date 2008/06/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Knot theory is the study of isotopy classes of embeddings of the circle S1 into a 3-manifold, specifically R3. The Fáry-Milnor Theorem says that any curve in 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 Γ, what limitations on the isotopy class of Γ are implied by a bound on total curvature? What does ``total curvature" mean for a graph? We define a natural notion of net total curvature of a graph Γ in R3, and prove that if Γ is homeomorphic to the θ-graph, then the net total curvature of Γ ≥ 3π; and if it is < 4π, then Γ is isotopic in R3 to a planar θ-graph. Further, the net total curvature = 3π only when Γ is a convex plane curve plus a chord. We begin our discussion with piecewise smooth graphs, and extend all these results to continuous graphs in the final section. In particular, we show that continuous graphs of finite total curvature are isotopic to polygonal graphs.