Minimally knotted graphs in \(S^ 3\)
type: Item
instance of: scholarly article
title: Minimally knotted graphs in \(S^ 3\)
zbMATH Open document ID: 0723.57008
DOI: 10.1016/0166-8641(90)90061-6
author: Jonathan K. Simon
author: Keith Wolcott
published in: Topology and its Applications
publication date: 1990-01-01T00:00:00Z
review text: Let G be a planar graph embedded in \(S^ 3\). If the embedding is planar, then G is said to be unknotted. If each proper subgraph of G is unknotted, then G is said to be locally unknotted. It is natural to conjecture that if \(\pi_ 1(S^ 3-G)\) is free and G is locally unknotted, then G is unknotted. In the first half of this paper, the authors prove the above conjecture for various kind of graphs including handcuff graphs, \(\theta\)-curves, and double \(\theta\)-curves. The proof uses some results of combinatorial group theory. Since then, Scharlemann and Thompson proved the above conjecture in full generality. In the latter half of this paper, the authors give a new method for proving nontriviality of a trivalent graph, by developing an algorithm that starts with a trivalent graph G and produces a trivalent graph \(G_ I\) with fewer vertices than G such that \(G_ I\) nonplanar embedded implies G nonplanar embedded. By this method, they reprove the nontriviality of several minimally knotted graphs (that is, locally unknotted graphs which are nontrivial). It is noted that \textit{A. Kawauchi} [Osaka J. Math. 26, 743-758 (1989; Zbl 0701.57015)] proved their conjecture that every planar graph has a minimally knotted embedding.
reviewed by: Makoto Sakuma
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M15
zbMATH DE Number: 4192660
zbMATH Keywords: spatial graphs
zbMATH Keywords: planar graph
zbMATH Keywords: handcuff graphs
zbMATH Keywords: \(\theta \) -curves
zbMATH Keywords: minimally knotted graphs
MaRDI profile type: Publication
full work available at URL: 0166-8641(90)90061-6
OpenAlex ID: W1971811830
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description: scientific article; zbMATH DE number 4192660