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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 3(405), Pages 63–114 (Mi umn9476)  

This article is cited in 4 scientific papers (total in 5 papers)

Roots and decompositions of three-dimensional topological objects

S. V. Matveevab

a Chelyabinsk State University
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

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 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.

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

DOI: https://doi.org/10.4213/rm9476

Full text: PDF file (962 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2012, 67:3, 459–507

Bibliographic databases:

Document Type: Article
UDC: 515.162.3
MSC: 57M25, 57M27
Received: 03.10.2011

Citation: S. V. Matveev, “Roots and decompositions of three-dimensional topological objects”, Uspekhi Mat. Nauk, 67:3(405) (2012), 63–114; Russian Math. Surveys, 67:3 (2012), 459–507

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    This publication is cited in the following articles:
    1. N. Dolbilin, G. Edelsbrunner, A. Ivanov, O. Musin, “Pervaya yaroslavskaya letnyaya shkola po diskretnoi i vychislitelnoi geometrii”, Model. i analiz inform. sistem, 19:4 (2012), 168–173  mathnet
    2. A. M. Kulakova, “Dokazatelstvo teoremy Shuberta”, Vestnik ChelGU, 2013, no. 16, 125–129  mathnet
    3. Meili Zhang, Bo Deng, “Prime decomposition of three-dimensional manifolds into boundary connected sum”, ISRN Appl. Math., 2014 (2014), 717265, 3 pp.  crossref  mathscinet  zmath  adsnasa
    4. Ph. G. Korablev, Ya. K. May, “Knotoids and knots in the thickened torus”, Siberian Math. J., 58:5 (2017), 837–844  mathnet  crossref  crossref  isi  elib  elib
    5. V. M. Buchstaber, V. A. Vassiliev, A. Yu. Vesnin, I. A. Dynnikov, Yu. G. Reshetnyak, A. B. Sossinsky, I. A. Taimanov, V. G. Turaev, A. T. Fomenko, E. A. Fominykh, A. V. Chernavsky, “Sergei Vladimirovich Matveev (on his 70th birthday)”, Russian Math. Surveys, 73:4 (2018), 737–746  mathnet  crossref  crossref  adsnasa  isi  elib
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