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Invariants of graphs in three-space. (English) Zbl 0672.57008
Motivated by chemistry and molecular biology, the topological study of isotopy classes of finite graphs embedded in 3-dimensional space (as an extension of knot theory) has received interest over the past few years. The author develops invariants for such embedded graphs. The general principle is the following: Given such an embedded graph, ‘surgering’ its vertices provides a family of disjointly embedded arcs and circles in 3- space; in particular, these circles form a link in 3-space. There are of course several possible such surgical procedures, each giving a link. However, considering all possible surgeries, the list of all links obtained in this way forms a topological invariant of the embedded graph. This procedure turns out to be particularly efficient for rigid vertex graphs. These are graphs endowed with an additional structure near each vertex, requiring that there is a plane passing through this vertex which contains the beginning of all edges emanating from this vertex. In this case, it is possible to restrict the type of surgical procedures allowed so that they give only links, without embedded arcs. The author shows how to use this technique to distinguish some embedded graphs from their mirror image.
Reviewer: F.Bonahon

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q35 Embeddings and immersions in PL-topology
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