Roots and decompositions of three-dimensional topological objects

S. V. Matveev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 3, pp. 459-507

Résumé

In 1942 M. H. A. Newman formulated and proved a simple lemma of great importance for various fields of mathematics, including algebra and the theory of Gröbner–Shirshov bases. Later it was called the Diamond Lemma, since its key construction was illustrated by a diamond-shaped diagram. In 2005 the author suggested a new version of this lemma suitable for topological applications. This paper gives a survey of results on the existence and uniqueness of prime decompositions of various topological objects: three-dimensional manifolds, knots in thickened surfaces, knotted graphs, three-dimensional orbifolds, and knotted theta-curves in three-dimensional manifolds. As it turned out, all these topological objects admit a prime decomposition, although it is not unique in some cases (for example, in the case of orbifolds). For theta-curves and knots of geometric degree 1 in a thickened torus, the algebraic structure of the corresponding semigroups can be completely described. In both cases the semigroups are quotients of free groups by explicit commutation relations. Bibliography: 33 titles.

Détail

Keywords: three-dimensional manifold, knot, virtual knot, orbifold.
Mots-clés : prime decomposition

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