Geodesic
Virtual $3$-manifolds of complexity $1$ and~$2$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 257-264

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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, complexity.
Mots-clés : classification 
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     author = {E. A. Sbrodova and V. V. Tarkaev and E. A. Fominykh and E. V. Shumakova},
     title = {Virtual $3$-manifolds of complexity $1$ and~$2$},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {257--264},
     publisher = {mathdoc},
     volume = {23},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a24/}
}
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E. A. Sbrodova; V. V. Tarkaev; E. A. Fominykh; E. V. Shumakova. Virtual $3$-manifolds of complexity $1$ and~$2$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 257-264. http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a24/