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Basic embeddings of graphs and the method of three-page Dynnikov embeddings. (English. Russian original) Zbl 1054.05029

Russ. Math. Surv. 58, No. 2, 372-374 (2003); translation from Usp. Mat. Nauk 58, No. 2, 163-164 (2003).
Summary: This note is devoted to the solution of two problems of contemporary geometric topology which have a common method of solution – the technique of embedding graphs in a book with finitely many pages. First, we prove a criterion for the basic embeddability of finite graphs in \(\mathbb{R}\times T_{n,m}\), where \(T_{n,m}\) is a bouquet of \(n\) segments (glued by their ends) and \(m\) circles. Second, the isotopy classification of arbitrary knotted graphs is reduced to the algebraic problem of the equality of central elements in two series of finitely presented semigroups.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
57M15 Relations of low-dimensional topology with graph theory
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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