Research output: Contribution to journal › Article › peer-review
Virtual 3-Manifolds of Complexity 1 and 2. / Sbrodova, E. A.; Tarkaev, V. V.; Fominykh, E. A.; Shumakova, E. V.
In: Proceedings of the Steklov Institute of Mathematics, Vol. 304, 01.04.2019, p. S154-S160.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Virtual 3-Manifolds of Complexity 1 and 2
AU - Sbrodova, E. A.
AU - Tarkaev, V. V.
AU - Fominykh, E. A.
AU - Shumakova, E. V.
PY - 2019/4/1
Y1 - 2019/4/1
N2 - Matveev in 2009 introduced the notion of virtual 3-manifold, which generalizes the classical notion of 3-manifold. A virtual 3-manifold is an equivalence class of so-called special polyhedra. Each virtual 3-manifold determines a 3-manifold with nonempty boundary and ℝP2-singularities. Many invariants of manifolds, such as Turaev–Viro invariants, can be extended to virtual 3-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 fewer 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.
AB - Matveev in 2009 introduced the notion of virtual 3-manifold, which generalizes the classical notion of 3-manifold. A virtual 3-manifold is an equivalence class of so-called special polyhedra. Each virtual 3-manifold determines a 3-manifold with nonempty boundary and ℝP2-singularities. Many invariants of manifolds, such as Turaev–Viro invariants, can be extended to virtual 3-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 fewer 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.
KW - classification
KW - complexity
KW - virtual 3-manifold
UR - http://www.scopus.com/inward/record.url?scp=85067083227&partnerID=8YFLogxK
U2 - 10.1134/S0081543819020172
DO - 10.1134/S0081543819020172
M3 - Article
AN - SCOPUS:85067083227
VL - 304
SP - S154-S160
JO - Proceedings of the Steklov Institute of Mathematics
JF - Proceedings of the Steklov Institute of Mathematics
SN - 0081-5438
ER -
ID: 49856293