Graphical neighborhoods of spatial graphs
type: Item
instance of: scholarly article
title: Graphical neighborhoods of spatial graphs
zbMATH Open document ID: 1490.57028
DOI: 10.1007/978-3-030-62497-2_38
author: Stefan Friedl
author: Gerrit Herrmann
publication date: 2021-12-10T00:00:00Z
review text: In classical knot theory, extensive use is made of the fact that a knot in the 3-sphere can be thickened to a tubular neighborhood and, for many purposes, the properties of the tubular neighborhood are interchangeable with those of the knot. Furthermore, it does not matter if one works in the piecewise-linear setting or the smooth setting. In this paper, the authors develop the analogous theory for spatial graphs in the smooth setting. A ``graphical neighborhood'' is the spatial graph version of the tubular neighborhood of a knot. Although spatial graphs have been extensively studied, previously no one has carefully developed the foundational topological theory. The main results of this paper include careful definitions of spatial graph and graphical neighborhood, the existence and uniqueness (up to a certain equivalence) of graphical neighborhoods, and the fact that \(S^3 \setminus N(G)\) is a deformation retract \(S^3 \setminus G\), where \(G\) is a spatial graph and \(N(G)\) is a graphical neighborhood. The main challenge of the proofs of each theorem lies in how vertices of degree 3 or greater are treated, as differential topology is classically concerned with manifolds and spatial graphs are not manifolds. For the entire collection see [Zbl 1459.37002].
reviewed by: Scott A. Taylor
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57K10
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 7442381
zbMATH Keywords: knot
zbMATH Keywords: spatial graph
zbMATH Keywords: tubular neighborhood
zbMATH Keywords: graphical neighborhood
MaRDI profile type: Publication
full work available at URL: 978-3-030-62497-2 38
OpenAlex ID: W3129762076
cites work: Q5000901
cites work: Q5614468
cites work: Q4114528
cites work: Q4779969
cites work: Q4658185
cites work: Q5667781
cites work: On irreducible 3-manifolds which are sufficiently large
cites work: Differential Topology
description: scientific article; zbMATH DE number 7442381