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- W2059632904 abstract "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 Grobner-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." @default.
- W2059632904 created "2016-06-24" @default.
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- W2059632904 date "2012-06-30" @default.
- W2059632904 modified "2023-10-14" @default.
- W2059632904 title "Roots and decompositions of three-dimensional topological objects" @default.
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- W2059632904 doi "https://doi.org/10.1070/rm2012v067n03abeh004794" @default.
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