Papers by Evgeny Fominykh
Uspekhi Matematicheskikh Nauk
Uspekhi Matematicheskikh Nauk
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored ... more The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric boundary (up to graph complexity 12) and for infinite families of lens spaces.
Results in Mathematics
A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored grap... more A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e. 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere.
Doklady Mathematics, 2002
Sbornik: Mathematics, 2016
Virtual 3-manifolds were introduced by S.V. Matveev in 2009 as natural generalizations of the cla... more Virtual 3-manifolds were introduced by S.V. Matveev in 2009 as natural generalizations of the classical 3-manifolds. In this paper, we introduce a notion of complexity of a virtual 3-manifold. We investigate the values of the complexity for virtual 3-manifolds presented by special polyhedra with one or two 2-components. On the basis of these results, we establish the exact values of the complexity for a wide class of hyperbolic 3-manifolds with totally geodesic boundary.
Siberian Mathematical Journal, 2012
The nonintersecting classes H p,q are defined, with p, q ∈ N and p ≥ q ≥ 1, of orientable hyperbo... more The nonintersecting classes H p,q are defined, with p, q ∈ N and p ≥ q ≥ 1, of orientable hyperbolic 3-manifolds with geodesic boundary. If M ∈ H p,q , then the complexity c(M ) and the Euler characteristic χ(M ) of M are related by the formula c(M ) = p−χ(M ). The classes H q,q , q ≥ 1, and H 2,1 are known to contain infinite series of manifolds for each of which the exact values of complexity were found. There is given an infinite series of manifolds from H 3,1 and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the ε-invariants of manifolds.
Proceedings of the Steklov Institute of Mathematics
A special spine of a three-manifold is said to be poor if it does not contain proper simple subpo... more A special spine of a three-manifold is said to be poor if it does not contain proper simple subpolyhedra. Using the Turaev-Viro invariants, we establish that every compact three-dimensional manifold M with connected nonempty boundary has a finite number of poor special spines. Moreover, all poor special spines of the manifold M have the same number of true vertices. We prove that the complexity of a compact hyperbolic three-dimensional manifold with totally geodesic boundary that has a poor special spine with two 2-components and n true vertices is n. Such manifolds are constructed for infinitely many values of n.
Siberian Advances in Mathematics
Journal of differential geometry
Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both no... more Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) almost normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results: 1) a minimal triangulation of a closed irreducible or a bounded hyperbolic 3-manifold contains no non-trivial k-normal sphere; 2) every triangulation of a closed manifold with at least 2 tetrahedra contains some non-trivial normal surface; 3) every manifold with boundary has only finitely many triangulations without non-trivial normal surfaces. Here, triangulations of bounded manifolds are actually ideal triangulations. We also calculate the number of normal surfaces of nonnegative Euler characteristics which are contained in the conjecturally minimal triangulations of all lens spaces.
ABSTRACT The set of all normal surfaces in a 3-manifold is a partial monoid under addition with a... more ABSTRACT The set of all normal surfaces in a 3-manifold is a partial monoid under addition with a minimal generating set of fundamental surfaces. The available algorithm for finding the system of fundamental surfaces is of a theoretical nature and admits no implementation in practice. In this article, we give a complete and geometrically simple description for the structure of partial monoids for normal surfaces in lens spaces, generalized quaternion spaces, and Stallings manifolds with fiber a punctured torus and a hyperbolic monodromy map.
Siberian Mathematical Journal, 2011
We establish an upper bound ω(p/q) on the complexity of manifolds obtained by p/q-surgeries on th... more We establish an upper bound ω(p/q) on the complexity of manifolds obtained by p/q-surgeries on the figure eight knot. It turns out that if ω(p/q) 12, the bound is sharp.
Journal of Knot Theory and Its Ramifications, 2013
We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary.... more We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements. 57M20, 57M25; 57M50 arXiv:1302.3863v1 [math.GT] 15 Feb 2013 2 Seifert fibered manifolds with boundary
Сибирские электронные …, 2005
Abstract: We provide a new formula for an upper bound of the complexity of non-Seifert graph-mani... more Abstract: We provide a new formula for an upper bound of the complexity of non-Seifert graph-manifolds obtained by gluing together two Seifert manifolds fibered over the disc with two exceptional fibers. This bound turns out to be sharp for many manifolds.
Doklady Mathematics, 2014
We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tet... more We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21 (non-orientable case) tetrahedra. Our isometry classification uses certified canonical cell decompositions (based on work by Dunfield, Hoffman, Licata) and isomorphism signatures (an improvement of dehydration sequences by Burton). The tetrahedral census comes in Regina as well as SnapPy format, and we illustrate its features.
Doklady Mathematics, 2011
There are found exact values of (Matveev) complexity for the 2-parameter family of hyperbolic 3-m... more There are found exact values of (Matveev) complexity for the 2-parameter family of hyperbolic 3-manifolds with boundary constructed by Paoluzzi and Zimmermann. Moreover, ε-invariants for these manifolds are calculated.
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Papers by Evgeny Fominykh