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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2017, Volume 23, Number 4, Pages 257–264
DOI: https://doi.org/10.21538/0134-4889-2017-23-4-257-264
(Mi timm1485)
 

Virtual $3$-manifolds of complexity $1$ and $2$

E. A. Sbrodovaa, V. V. Tarkaevba, E. A. Fominykhba, E. V. Shumakovaa

a Chelyabinsk State University, Chelyabinsk, 454001 Russia
b Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia
References:
Abstract: Matveev in 2009 introduced the notion of virtual $3$-manifold, which generalizes the classical notion of $3$-manifold. A virtual manifold is an equivalence class of so-called special polyhedra. Each virtual manifold determines a $3$-manifold with nonempty boundary and $\mathbb{R}P^2$-singularities. Many invariants of manifolds, such as Turaev–Viro invariants, can be extended to virtual manifolds. The complexity of a virtual $3$-manifold is $k$ if its equivalence class contains a special polyhedron with $k$ true vertices and contains no special polyhedra with a smaller number of true vertices. In this paper we give a complete list of virtual $3$-manifolds of complexity $1$ and present two-sided bounds for the number of virtual $3$-manifolds of complexity $2$. The question of the complete classification for virtual $3$-manifolds of complexity $2$ remains open.
Keywords: virtual $3$-manifold, classification, complexity.
Funding agency Grant number
Russian Foundation for Basic Research 16-01-00609
Received: 30.09.2017
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2019, Volume 304, Issue 1, Pages S154–S160
DOI: https://doi.org/10.1134/S0081543819020172
Bibliographic databases:
Document Type: Article
UDC: 515.162
MSC: 57N10, 57M27
Language: Russian
Citation: E. A. Sbrodova, V. V. Tarkaev, E. A. Fominykh, E. V. Shumakova, “Virtual $3$-manifolds of complexity $1$ and $2$”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 4, 2017, 257–264; Proc. Steklov Inst. Math. (Suppl.), 304, suppl. 1 (2019), S154–S160
Citation in format AMSBIB
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\paper Virtual $3$-manifolds of complexity $1$ and~$2$
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 4
\pages 257--264
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\crossref{https://doi.org/10.21538/0134-4889-2017-23-4-257-264}
\elib{https://elibrary.ru/item.asp?id=30713979}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2019
\vol 304
\issue , suppl. 1
\pages S154--S160
\crossref{https://doi.org/10.1134/S0081543819020172}
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