Homology classification of spatial graphs by linking numbers and Simon invariants
type: Item
instance of: scholarly article
title: Homology classification of spatial graphs by linking numbers and Simon invariants
zbMATH Open document ID: 1032.57002
DOI: 10.1016/S0166-8641(03)00101-9
author: Reiko Shinjo
author: Kouki Taniyama
published in: Topology and its Applications
publication date: 2003-10-22T00:00:00Z
review text: Spatial-graph-homology is an equivalence relation on spatial graphs introduced by the second author [Topology 33, 509-523 (1994; Zbl 0823.57006)] as a generalization of link-homology [\textit{H. Murakami} and \textit{Y. Nakanishi}, Math. Ann. 284, 75-89 (1989; Zbl 0646.57005)]. He showed in another paper [Topology Appl. 65, 205-228 (1995; Zbl 0843.57012)] that two spatial embeddings of a graph are spatial-graph-homologous if and only if they have the same Wu invariant [\textit{W. Wu}, Sci. Sin. 9, 21-46 (1960; Zbl 0121.18102)]. \textit{A. Yasuara} also gave a homology classification of spatial graphs by using disk/band surfaces [Topology Appl. 69, 173-191 (1996; Zbl 0860.57004)]. In the paper under review, the authors give another homology classification of spatial graphs in terms of the linking number and the Simon invariant, where the Simon invariant is an invariant of spatial embeddings of the complete graph \(K_{5}\) on five vertices and the complete bipartite graph \(K_{3,3}\) on \(3+3\) vertices. It should be noted that the Wu invariant of a spatial embedding of a graph \(G\) coincides with the linking number if \(G\) is homeomorphic to the disjoint union of two \(1\)-spheres and the Simon invariant if \(G\) is homeomorphic to \(K_{5}\) or \(K_{3,3}\). The authors show that two spatial embeddings \(f\) and \(g\) of a graph \(G\) have the same Wu invariant if and only if for any subgraph \(H\) of \(G\) that is homeomorphic to the disjoint union of two \(1\)-spheres, \(K_{5}\) or \(K_{3,3}\), the restriction embeddings \(f|_{H}\) and \(g|_{H}\) have the same Wu invariant. Namely, the spatial-graph-homology class of a spatial embedding of a graph can be determined by linking numbers and Simon invariants. This is a generalization of the result shown in [\textit{T. Soma, H. Sugai} and \textit{A. Yasuhara}, Tokyo J. Math. 20, 1-11 (1997; Zbl 0891.05021)].
reviewed by: Ryo Nikkuni
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 1995438
zbMATH Keywords: spatial graph
zbMATH Keywords: spatial-graph-homology
zbMATH Keywords: delta move
zbMATH Keywords: linking number
zbMATH Keywords: Simon invariant
zbMATH Keywords: finite type invariant
MaRDI profile type: Publication
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full work available at URL: s0166-8641(03)00101-9
OpenAlex ID: W1985375787
description: scientific article; zbMATH DE number 1995438