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.
The paper was received by the Editorial Office on September 30, 2017
Elena Aleksandrovna Sbrodova, Cand. Sci.(Phys.-Math.), Chelyabinsk State University, Chelyabinsk,
454001 Russia, e-mail: sbrodova@csu.ru .
Vladimir Viktorovich Tarkaev, Cand. Sci.(Phys.-Math.), Krasovskii Institute of Mathematics and
Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia;
Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: v.tarkaev@gmail.com.
Evgeny Anatol’evich Fominykh, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics
and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia;
Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: efominykh@gmail.com.
Ekaterina Valer’evna Shumakova, Chelyabinsk State University, Chelyabinsk, 454001 Russia,
e-mail: shumakova_kate@mail.ru .
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