Virtual 3-Manifolds of Complexity 1 and 2

E. A. Sbrodova, V. V. Tarkaev, E. A. Fominykh, E. V. Shumakova

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish
Pages (from-to)S154-S160
Number of pages7
JournalProceedings of the Steklov Institute of Mathematics
StatePublished - 1 Apr 2019

Scopus subject areas

  • Mathematics (miscellaneous)


  • classification
  • complexity
  • virtual 3-manifold


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