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M. Skopenkov

Publications and source records attributed to M. Skopenkov.

13 recordsLinked to original sources
arXiv2026

Approximation by Quad Meshes in Laguerre Geometry

We study analogs of planar-quadrilateral meshes in Laguerre sphere geometry and the approximation of smooth surfaces by them. These new Laguerre meshes can be viewed as watertight surfaces formed by planar quadrilaterals (corresponding to the vertices of a mesh), strips of right circular cones (representing the edges), and spherical faces. In the smooth limit, we get an analog of conjugate nets in Laguerre geometry, which we call Laguerre conjugate nets with respect to an attached sphere congruence. We introduce the notion of Laguerre conjugate directions, provide a method for computing them, and apply them to approximate surfaces by L-meshes with prescribed radii of spherical faces.

cs.CG↗
arXiv2025

Quasi-symmetric nets: A constructive approach to the equimodular elliptic type of Kokotsakis polyhedra

A Kokotsakis polyhedron is a polyhedral mesh in three-dimensional Euclidean space formed by a central n-gonal face (the base), n quadrilateral faces each sharing one edge with the base, and n triangular faces inserted between every two adjacent quadrilaterals; it is called flexible if it admits a continuous deformation that preserves the rigidity of every face. This work investigates flexible Kokotsakis polyhedra with a quadrangular base (n = 4) of equimodular elliptic type, filling a significant gap in the literature by providing the first explicit constructions of this type together with an explicit algebraic characterization in terms of flat and dihedral angles. A straightforwardly constructible class of polyhedra - called quasi-symmetric nets (QS-nets) - is introduced, characterized by a symmetry relation among flat angles. It is shown that every elliptic QS-net has equimodular elliptic type and is flexible in real three-dimensional Euclidean space (rather than only in complex configuration spaces), except for a few exceptional choices of dihedral angles, and that its flexion admits a closed-form parameterization. Examples are constructed that are non-self-intersecting and belong exclusively to the equimodular elliptic type. To support applications in computational geometry, a numerical pipeline is developed that searches for candidate solutions, verifies them using the explicit algebraic characterization, and constructs and visualizes the resulting polyhedra; numerical validations achieve high precision. Taken together, these results provide constructive criteria, algorithms, and validated examples for the equimodular elliptic type, enabling the design of a broad range of flexible Kokotsakis mechanisms.

math.MG↗
arXiv2025

Incidences, tilings, and fields

The master theorem, introduced by Richter-Gebert and generalized by Fomin and the first author, provides a method for proving incidence theorems of projective geometry using triangular tilings of surfaces. We investigate which incidence theorems over C and R can or cannot be proved via the master theorem. For this, we formalize the notion of a tiling proof. We introduce a hierarchy of classes of theorems based on the underlying topological spaces. A key tool is considering the same theorems over finite fields.

math.CO↗
arXiv2025

Flexible polyhedral nets in isotropic geometry

We study flexible polyhedral nets in isotropic geometry. This geometry has a degenerate metric, but there is a natural notion of flexibility. We study infinitesimal and finite flexibility, and classify all finitely flexible polyhedral nets of arbitrary size. We show that there are just two classes, in contrast to Izmestiev's rather involved classification in Euclidean geometry, for size 3x3 only. Using these nets to initialize the optimization algorithms, we turn them into approximate Euclidean mechanisms. We also explore the smooth versions of these classes.

math.MG↗
arXiv2024

Area preserving Combescure transformations

Motivated by the design of flexible nets, we classify all nets of arbitrary size m x n that admit a continuous family of area-preserving Combescure transformations. There are just two different classes. The nets in the first class are special cases of cone nets that have been recently studied by Kilian, Mueller, and Tervooren. The second class consists of Koenigs nets having a Christoffel dual with the same areas of corresponding faces. We apply isotropic metric duality to get a new class of flexible nets in isotropic geometry. We also study the smooth analogs of the introduced classes.

math.MG↗
arXiv2020

Feynman checkers: towards algorithmic quantum theory

We survey and develop the most elementary model of electron motion introduced by R$.$Feynman. In this game, a checker moves on a checkerboard by simple rules, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk or an Ising model at imaginary temperature. We solve mathematically a problem by R$.$Feynman from 1965, which was to prove that the discrete model (for large time, small average velocity, and small lattice step) is consistent with the continuum one. We study asymptotic properties of the model (for small lattice step and large time) improving the results by J$.$Narlikar from 1972 and by T$.$Sunada-T$.$Tate from 2012. For the first time we observe and prove concentration of measure in the small-lattice-step limit. We perform the second quantization of the model.

math-ph↗
arXiv2019

Elements of mathematics in problems. Through olympiads and circles to profession

This is a collection of teaching materials used in several Russian universities, schools, and mathematical circles. Most problems are chosen in such a way that in the course of the solution and discussion a reader learns important mathematical ideas and theories. The materials can be used by pupils and students for self-study, and by teachers. This is an abridged pre-copyedit version of the published book submitted with the permission of the publisher. Each included individual material is self-contained and ready-for-use. Solutions to problems are not included intentionally. This collection consolidates updates of several arXiv submissions, e.g., arXiv:1305.2598.

math.HO↗
arXiv2016

Tiling by rectangles and alternating current

This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings due to R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of the paper is an application of alternating-current circuits. The following results are obtained: - a necessary condition for a rectangle to be tilable by rectangles of given shapes; - a criterion for a rectangle to be tilable by rectangles similar to it but not all homothetic to it; - a criterion for a generic polygon to be tilable by squares. These results generalize the ones of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne and G. Szekeres.

math.CO↗
arXiv2015

Surfaces containing two circles through each point and Pythagorean 6-tuples

We study analytic surfaces in 3-dimensional Euclidean space containing two circular arcs through each point. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. We reduce finding all such surfaces to the algebraic problem of finding all Pythagorean 6-tuples of polynomials. The reduction is based on the Schicho parametrization of surfaces containing two conics through each point and a new approach using quaternionic rational parametrization.

math.AG↗
arXiv2008

Classification of knotted tori in the 2-metastable dimension

This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings $N\to S^m$. We study the specific case of knotted tori, i. e. the embeddings $S^p \times S^q \to S^m$. The classification of knotted tori up to isotopy in the metastable dimension range $m>p+\frac{3}{2}q+2$, $p\le q$, was given by A. Haefliger, E. Zeeman and A. Skopenkov. We consider the dimensions below the metastable range, and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension: Theorem. Assume that $p+\frac{4}{3}q+2 2p+q+2$. Then the set of smooth embeddings $S^p \times S^q \to S^m$ up to isotopy is infinite if and only if either $q+1$ or $p+q+1$ is divisible by 4. Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new $\beta$-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement.

math.GT↗
arXiv2008

Homotopy type of the complement of an immersion and classification of embeddings of tori

This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range $m\ge p+3q/2+2$, $p\le q$, which is a natural limit for the classical methods of embedding theory, has been explicitely described earlier. The aim of this note is to present an approach which allows for results in lower dimension.

math.GT↗
arXiv2007

Classification of framed links in 3-manifolds

We present a short proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in details: {\bf Theorem.} Let $M$ be a connected oriented closed smooth 3-manifold. Let $L_1(M)$ be the set of framed links in $M$ up to a framed cobordism. Let $°:L_1(M)\to H_1(M;\Z)$ be the map taking a framed link to its homology class. Then for each $α\in H_1(M;\Z)$ there is a 1-1 correspondence between the set $°\nolimits^{-1}α$ and the group $\Bbb Z_{2d(α)}$, where $d(α)$ is the divisibility of the projection of $α$ to the free part of $H_1(M;\Bbb Z)$.

math.GT↗