Homology classification of spatial graphs by linking numbers and Simon invariants
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)].
scientific article; zbMATH DE number 1995438