arXiv subjects
Publications and source records attributed to A. Skopenkov.
We recall definitions of linking numbers and Wu--Simon numbers for spatial graphs. We expose a `converse' to the Conway--Gordon--Sachs theorem (i.e. description of linking functions for embeddings $K_6\to\mathbb{R}^3$), and some results on Wu--Simon numbers. We conjecture and discuss a generalization of the Conway--Gordon--Sachs theorem to multiple linking. The exposition is based on plane diagrams, so no knowledge of spatial geometry is required.
In this expository note we present an elementary direct rigorous definition and the simplest properties of the winding number. This definition is simpler than the one given in some textbooks. We show how to compute the winding number easily: using additivity or counting the (signed) intersection points. In the language of the winding number, we present an elementary formulation and proof of the low-dimensional case of the Borsuk--Ulam theorem. An English version is followed by a Russian version.
This paper shares some experience in advanced mathematical education. We show how a high school student can be naturally and gradually introduced to basic steps of scientific research: developing intuition by finding and correcting mistakes through discussions and writing a paper, (transparent) anonymous peer review, recognition and award. We show that most of this can be done in research projects not aiming at scientific novelty. We share the experience (both principles and examples) of the Moscow Mathematical Conference of High School Students.
In this text we expose basic cases of some fundamental ideas and methods of topology. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal lines in a subset of the plane. Although these ideas and methods are parts of topology, they are used in other areas including computer science. This text is expository and is accessible to mathematicians not specialized in the area (and to students). The English version mostly consists of results and problems, and is followed by a more narrative Russian version having a different set of authors.
The Netflix problem (from machine learning) asks the following. Given a ratings matrix in which each entry $(i,j)$ represents the rating of movie $j$ by customer $i$, if customer $i$ has watched movie $j$, and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. The remaining entries are predicted so as to minimize the {\it rank} of the completed matrix. In this survey we study a more general problem, in which instead of knowing specific matrix elements, we know linear relations on such elements. We describe applications of these results to embeddings of graphs in surfaces (more precisely, embeddings with rotation systems, and embeddings modulo 2).
A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in combinatorial geometry, in topological combinatorics, and in studies of embeddings. We prove some relations between the invariants. We demonstrate the connection of some of these relations to homology of the deleted product of a graph. We construct almost embeddings realizing some values of these invariants. We present some ideas of algebraic and geometric topology in a language accessible to non-topologists (in particular, to students). All the necessary definitions are recalled. However elementary, this paper is motivated by frontline of research; there are some conjectures and open problems.
A graph drawing in the plane is called an almost embedding if images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. We introduce integer invariants of almost embeddings: winding number, cyclic and triodic Wu numbers. We construct almost embeddings realizing some values of these invariants. We prove some relations between the invariants. We study values realizable as invariants of some almost embedding, but not of any embedding. This paper is expository and is accessible to mathematicians not specialized in the area (and to students). However elementary, this paper is motivated by frontline of research.
In this expository paper we present some ideas of algebraic topology (more precisely, of homology theory) in a language accessible to non-specialists in the area. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is contained in an even number of edges from $C$. It is easy to check that the sum (modulo $2$) of $1$-cycles is a $1$-cycle. We start from the following problems: to find $\bullet$ the number of all $1$-cycles in a given graph; $\bullet$ a small number of $1$-cycles in a given graph such that any $1$-cycle is the sum of some of them. We consider generalizations (of these problems) to graphs with symmetry, to $2$-cycles in $2$-dimensional hypergraphs, and to certain configuration spaces of graphs (namely, to the square and the deleted square).
Let $f:S^q\sqcup S^q\to S^m$ be a link (i.e. an embedding). How does (the isotopy class of) the knot $S^q\to S^m$ obtained by embedded connected sum of the components of $f$ depend on $f$? Define a link $\sigma f:S^q\sqcup S^q\to S^m$ as follows. The first component of $\sigma f$ is the `standardly shifted' first component of $f$. The second component of $\sigma f$ is the embedded connected sum of the components of $f$. How does (the isotopy class of) $\sigma f$ depend on $f$? How does (the isotopy class of) the link $S^q\sqcup S^q\to S^m$ obtained by embedded connected sum of the last two components of a link $g:S^q_1\sqcup S^q_2\sqcup S^q_3\to S^m$ depend on $g$? We give the answers for the `first non-trivial case' $q=4k-1$ and $m=6k$. The first answer was used by S. Avvakumov for classification of linked 3-manifolds in $S^6$.
In this expository note we show how combinatorial Nullstellensatz by N. Alon naturally appears in solutions of elementary problems. Simple ideas gradually and naturally appear in such solutions, thus bringing a reader to generalizations. The note is accessible to mathematicians not specialized in the area, and to students familiar with polynomials.
This is an expository paper. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is contained in an even number of edges from $C$. E.g., a cycle in the sense of graph theory is a $1$-cycle, but not vice versa. It is easy to check that the sum (modulo $2$) of $1$-cycles is a $1$-cycle. In this text we study the following problems: to find $\bullet$ the number of all 1-cycles in a given graph; $\bullet$ a small number of 1-cycles in a given graph such that any 1-cycle is the sum of some of them. We also consider generalizations (of these problems) to graphs with symmetry, and to $2$-cycles in $2$-dimensional hypergraphs.
A general position map $f:K\to M$ of a $k$-dimensional simplicial complex to a $2k$-dimensional manifold (for $k=1$, of a graph to a surface) is a $\mathbb Z_2$-embedding if $|f\sigma \cap f\tau|$ is even for any non-adjacent $k$-faces $\sigma,\tau$. We present criteria for $\mathbb Z_2$-embeddability of certain $k$-dimensional complex (for $k=1$, of any graph) to $2k$-dimensional manifolds. These criteria are $\bullet$ a `Kuratowski-type' version of the Fulek-Kyn\v{c}l-Bikeev criteria (for $k=1$), and $\bullet$ a converse to the Dzhenzher-Skopenkov necessary condition (for $k>1$). Our higher-dimensional criterion allows us to reduce the modulo 2 K\"uhnel problem on embeddings to a purely algebraic problem. Our proof is interplay between geometric topology, combinatorics and linear algebra. It is based on calculation of generators in the homology of certain configuration space (the deleted product) of certain complex (joinpower).
The purpose of this short note is to guide a reader to a reliable reference for the following result of S. Parsa: For any $k,l\ge2$ there exist simplicial complexes $K, L$ of dimensions $k,l$ such that $K$ does not embed into $\mathbb R^{2k}$, and $L$ does not embed into $\mathbb R^{2l}$, but the join $K*L$ embeds into $\mathbb R^{2(k+l+1)}$.
We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilbert's 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are continuous functions $φ_1,\ldots,φ_5 : [\,0, 1\,]\to [\,0,1\,]$ such that for any continuous function $f: [\,0,1\,]^2\to\mathbb R$ there is a continuous function $h: [\,0,3\,]\to\mathbb R$ such that for any $x,y\in [\,0, 1\,]$ we have $$f(x,y)=\sum\limits_{k=1}^5 h\left(φ_k(x)+\sqrt{2}\,φ_k(y)\right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.
The classical Heawood inequality states that if the complete graph $K_n$ on $n$ vertices is embeddable in the sphere with $g$ handles, then $g \ge\dfrac{(n-3)(n-4)}{12}$. A higher-dimensional analogue of the Heawood inequality is the K\"uhnel conjecture. In a simplified form it states that for every integer $k>0$ there is $c_k>0$ such that if the union of $k$-faces of $n$-simplex embeds into the connected sum of $g$ copies of the Cartesian product $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge c_k n^{k+1}$. For $k>1$ only linear estimates were known. We present a quadratic estimate $g\ge c_k n^2$. The proof is based on beautiful and fruitful interplay between geometric topology, combinatorics and linear algebra.
Let $M$ be a sphere with handles and holes, $f:M\to\mathbb R^3$ an embedding, and $H_1=H_1(M;\mathbb Z)$. We study a simple isotopy invariant of $f$, the Seifert bilinear form $L(f):H_1\times H_1\to\mathbb Z$. Let $\cap:H_1\times H_1\to\mathbb Z$ be the intersection form of $M$. Then the Seifert form is $\cap$-symmetric, i.e., $L(f)(β,γ)-L(f)(γ,β)=β\capγ$ for any $β,γ\in H_1$. If $M$ has non-empty boundary, then any $\cap$-symmetric bilinear form $H_1\times H_1\to\mathbb Z$ is realizable as $L(f)$ for some embedding $f$. We present a characterization of realizable forms for the torus $M$. The results are simple and presumably known in folklore. We present a simplified exposition accessible to non-specialists.
We exhibit relations between van Kampen-Flores, Conway-Gordon-Sachs and Radon theorems, by presenting direct proofs of some implications between them. The key idea is an interesting relation between the van Kampen and the Conway-Gordon-Sachs numbers for restrictions of a map of $(d+2)$-simplex to $\mathbb R^d$ to the $(d+1)$-face and to the $[d/2]$-skeleton.
The topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for any integers $r,d>1$ and any continuous map $f:Δ\to\mathbb R^d$ of the $(d+1)(r-1)$-dimensional simplex there are pairwise disjoint faces $σ_1,\ldots,σ_r\subsetΔ$ such that $f(σ_1)\cap \ldots \cap f(σ_r)\ne\emptyset$. The conjecture was proved for a prime power $r$. Recently counterexamples for other $r$ were found. Analogously, the $r$-fold van Kampen-Flores conjecture holds for a prime power $r$ but does not hold for other $r$. The arguments form a beautiful and fruitful interplay between combinatorics, algebra and topology. We present a simplified exposition accessible to non-specialists in the area. We also mention some recent developments and open problems.