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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
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.
Received: 30.09.2017
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
Linking options:
https://www.mathnet.ru/eng/timm1485 https://www.mathnet.ru/eng/timm/v23/i4/p257
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Abstract page: | 267 | Full-text PDF : | 70 | References: | 48 | First page: | 4 |
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