Spatial graphs with local knots
type: Item
instance of: scholarly article
title: Spatial graphs with local knots
zbMATH Open document ID: 1276.57008
author: Erica Flapan
author: Blake Mellor
author: Ramin Naimi
published in: Revista Matemática Complutense
publication date: 2013-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.0479
review text: In 1987, \textit{S. Suzuki} [``A prime decomposition theorem for a graph in the 3-sphere'', in: Topology and computer science, (Atami, 1986), 259--276, Kinokuniya, Tokyo, (1987)] extended a classic result of \textit{H. Schubert} [Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturw. Kl. 1949, No.3, 50 S. (1949; Zbl 0031.28602)] to spatial graphs by providing a prime decomposition theorem for graphs in \(S^3\). This paper continues in this vein and shows that every locally knotted edge of a 3-connected spatial graph has a ball meeting the graph in an arc that contains all the local knots of that edge. It is shown that this unknotting ball is unique up to an isotopy that setwise fixes the graph. This result is applied to the study of topological symmetry groups that preserve orientation of certain graphs in \(S^3\).
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 6204596
zbMATH Keywords: spatial graphs
zbMATH Keywords: local knots
zbMATH Keywords: topological symmetry groups
zbMATH Keywords: prime decomposition
reviewed by: Lew Ludwig
MaRDI profile type: Publication
OpenAlex ID: W3098970732
arXiv ID: 1010.0479
cites work: Topological symmetry groups of \(K_{4r+3}\)
cites work: RIGIDITY OF GRAPH SYMMETRIES IN THE 3-SPHERE
cites work: Complete graphs whose topological symmetry groups are polyhedral
cites work: Topological symmetry groups of graphs embedded in the 3-sphere
cites work: TOPOLOGICAL SYMMETRY GROUPS OF COMPLETE GRAPHS IN THE 3-SPHERE
cites work: Q3292138
cites work: Q3924842
cites work: Homotopy equivalences of 3-manifolds with boundaries
cites work: Q5788031
cites work: Topological chirality of certain molecules
cites work: Transformations of finite period. II
DOI: 10.1007/S13163-011-0072-9
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description: scientific article; zbMATH DE number 6204596