Notes on spatial representations of graphs
type: Item
instance of: scholarly article
title: Notes on spatial representations of graphs
zbMATH Open document ID: 0814.57007
author: Toshiki Endo
author: Takashi Otsuki
published in: Hokkaido Mathematical Journal
publication date: 1995-06-18T00:00:00Z
review text: A spatial representation \(\overline{G}\) of \(G\) is a tame imbedding of \(G\) into \(\mathbb{R}^ 3\); it is locally unknotted, with respect to cycle basis \(\{C_ 1,\dots,C_ n\}\) of \(H_ 1(G;Z)\), if there are internally disjoint disks \(D_ 1,\dots,D_ n\) in \(\mathbb{R}^ 3\) so that, for \(i = 1,\dots,n\), \(D_ i \cap \overline{G} = \partial D_ i = \overline{C}_ i\). \textit{K. Kobayashi} [Hokkaido Math. J. 21, No. 1, 117-140 (1992; Zbl 0765.57003)] showed that all complete graphs have locally unknotted spatial representations. The present authors extend this result to arbitrary graphs. They also show that any two minimal book presentations of a fixed complete graph are ambient isotopic in \(\mathbb{R}^ 3\) up to sheet translations, verifying a conjecture of Kobayashi.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 57M15
zbMATH DE Number: 720794
zbMATH Keywords: spatial representation
zbMATH Keywords: locally unknotted
zbMATH Keywords: minimal book presentations
zbMATH Keywords: complete graph
zbMATH Keywords: sheet translations
reviewed by: Arthur T. White
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381413094
OpenAlex ID: W2154731244
DOI: 10.14492/HOKMJ/1381413094
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description: scientific article; zbMATH DE number 720794