arXiv subjects
Publications and source records attributed to E. Alkin.
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.
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.
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.
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.