TY - JOUR

T1 - Total curvature of graphs in space

AU - Gulliver, Robert

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2007

Y1 - 2007

N2 - 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].

AB - 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].

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U2 - 10.4310/PAMQ.2007.v3.n3.a5

DO - 10.4310/PAMQ.2007.v3.n3.a5

M3 - Article

AN - SCOPUS:76849087145

SN - 1558-8599

VL - 3

SP - 773

EP - 783

JO - Pure and Applied Mathematics Quarterly

JF - Pure and Applied Mathematics Quarterly

IS - 3

ER -