arXiv subjects

A. Miroshnikov

Publications and source records attributed to A. Miroshnikov.

4 recordsLinked to original sources

arXiv2026

The winding number of a closed curve around a point

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.

math.HO↗

arXiv2024

Invariants of almost embeddings of graphs in the plane

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.

math.GT↗

arXiv2024

Invariants of almost embeddings of graphs in the plane: results and 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.

math.CO↗

arXiv2024

Cycles in graphs and in hypergraphs: towards homology theory

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).

math.HO↗