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Mikhail Skopenkov

Publications and source records attributed to Mikhail Skopenkov.

30 records · Page 2Linked to original sources
arXiv2016

Suspension theorems for links and link maps

We present a new short proof of the explicit formula for the group of links (and also link maps) in the 'quadruple point free' dimension. Denote by $L^m_{p,q}$ (respectively, $C^{m-p}_p$) the group of smooth embeddings $S^p\sqcup S^q\to S^m$ (respectively, $S^p\to S^m$) up to smooth isotopy. Denote by $LM^m_{p,q}$ the group of link maps $S^p\sqcup S^q\to S^m$ up to link homotopy. Theorem 1. If $p\le q\le m-3$ and $2p+2q\le 3m-6$ then \begin{equation*} L^m_{p,q}\cong π_p(S^{m-q-1})\oplusπ_{p+q+2-m}(SO/SO_{m-p-1})\oplus C^{m-p}_p\oplus C^{m-q}_q. \end{equation*} Theorem 2. If $p, q\le m-3$ and $2p+2q\le 3m-5$ then $LM^m_{p,q}\cong π^S_{p+q+1-m}$. Our approach is based on the use of the suspension operation for links and link maps, and suspension theorems for them.

math.GT↗
arXiv2015

When is the set of embeddings finite up to isotopy?

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddings N->S^m finite? In case when the manifold N is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold N is a disjoint union of spheres the answer was given by D. Crowley, S. Ferry and the author in 2011. We consider the next natural case when N is a product of two spheres. In the following theorem, FCS(i,j) is a concrete subset of the integer lattice depending only on the parity of i and j which is defined in the paper. Theorem. Assume that m>2p+q+2 and m S^m is infinite if and only if either q+1 or p+q+1 is divisible by 4, or there exists a point (x,y) in the set FCS(m-p-q,m-q) such that (m-p-q-2)x+(m-q-2)y=m-3. Our approach is based on a group structure on the set of embeddings and a new exact sequence, which in some sense reduces the classification of embeddings S^p x S^q -> S^m to the classification of embeddings S^{p+q} |_| S^q -> S^m and D^p x S^q -> S^m. The latter classification problems are reduced to homotopy ones, which are solved rationally.

math.GT↗
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↗
arXiv2013

Discrete Riemann surfaces: linear discretization and its convergence

We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also prove a discrete counterpart of the Riemann--Roch theorem. The proofs use energy estimates inspired by electrical networks.

math.CV↗
arXiv2013

On the Pontryagin-Steenrod-Wu theorem

This paper is on homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f of an (n+1)-manifold into the n-sphere define the degree deg f to be the class dual to f^*[S^n], where [S^n] is the fundamental class. We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem: Theorem. Let M be a connected orientable closed smooth (n+1)-manifold, n>2. Then the map deg:π^n(M)\to H_1(M;Z) is 1-to-1 (i.e., bijective), if the product w_2(M) x r_2 H_2(M;Z) is nonzero, where r_2 is the mod2 reduction; 2-to-1 (i.e., each element of H_1(M;Z) has exactly 2 preimages) - otherwise. The proof is based on the Pontryagin-Thom construction and a geometric definition of the Stiefel-Whitney classes w_2(M).

math.GT↗
arXiv2013

A square from similar rectangles

In the present popular science paper we determine when a square can be dissected into rectangles similar to a given rectangle. The approach to the question is based on a physical interpretation using electrical networks. Only secondary school background is assumed in the paper.

math.HO↗
arXiv2013

The boundary value problem for discrete analytic functions

This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal. We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant-K. Friedrichs-H. Lewy and L. Lusternik for square lattices, by D. Chelkak-S. Smirnov and implicitly by P.G. Ciarlet-P.-A. Raviart for rhombic lattices. In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory.

math.CO↗
arXiv2011

The rational classification of links of codimension >2

Fix an integer m and a multi-index p = (p_1, ..., p_r) of integers p_i < m-2. The set of links of codimension > 2, with multi-index p, E(p, m), is the set of smooth isotopy classes of smooth embeddings of the disjoint union of the p_i-spheres into the m-sphere. Haefliger showed that E(p, m) is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. r=1. For r > 1 and for restrictions on p the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group E(p, m) in general. In particular we determine precisely when E(p,m) is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras.

math.AT↗
arXiv2011

Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.

math.AG↗
arXiv2010

Packing a cake into a box

Given a cake in form of a triangle and a box that fits the mirror image of the cake, how to cut the cake into a minimal number of pieces so that it can be put into the box? The cake has an icing, so that we are not allowed to put it into the box upside down. V.G. Boltyansky asked this question in 1977 and showed that three pieces always suffice. In this paper we provide examples of cakes that cannot be cut into two pieces to put into the box. This shows that three is the answer to V.G. Boltyansky's question. Also we give examples of cakes which can be cut into two pieces.

math.CO↗
arXiv2008

A short proof of the Twelve points theorem

We present a short elementary proof of the following Twelve Points Theorem: Let M be a convex polygon with vertices at the lattice points, containing a single lattice point in its interior. Denote by m (resp. m*) the number of lattice points in the boundary of M (resp. in the boundary of the dual polygon). Then m+m*=12.

math.MG↗