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Roots and decompositions of three-dimensional topological objects. (English. Russian original) Zbl 1261.57007

Russ. Math. Surv. 67, No. 3, 459-507 (2012); translation from Usp. Mat. Nauk. 67, No. 3, 63-114 (2012).
In \(3\)-dimensional topology, various decomposition problems arise; the classical example is the unique decomposition of oriented closed \(3\)-manifolds by Kneser-Milnor. Along the lines of an old paper by M. H. A. Newman [Ann. Math. (2) 43, 223–243 (1942; Zbl 0060.12501)], the author develops a graph theoretical pattern which he applies to the existence and uniqueness question in decomposing the \(3\)-dimensional objects of the following list: prime decomposition of closed \(3\)-manifolds, for general compact \(3\)-manifolds with respect to boundary connected sum, knotted graphs in \(3\)-manifolds, orbifolds, annular reduction of \(3\)-manifolds, knots in thickened surfaces, virtual knots, \(\theta\)-curves, and cyclic decompositions of knots in a thickened torus. From the \(3\)-dimensional objects under study, the vertices of a directed graph are built; the edges are given by the appropriate reductions. Those ends of oriented paths in the graph which are sinks, the so-called roots, stand for the final result of decomposition. Unique roots are guaranteed by the diamond lemma, which requires the graph to fulfill a finiteness condition on oriented paths and presupposes the existence of a “mediator function” for pairs of edges emanating from the same vertex. Roots exist by the finiteness condition. The mediator function allows to compare the various edges emanating from a single vertex and leads to uniqueness of roots.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
05C20 Directed graphs (digraphs), tournaments
57M15 Relations of low-dimensional topology with graph theory

Citations:

Zbl 0060.12501
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