Invariants of almost embeddings of graphs in the plane

Abstract: In this survey we motivate studies of the invariants from the title. 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. We introduce integer invariants of almost embeddings: winding number, cyclic and triodic Wu numbers. We prove some relations between the invariants. We demonstrate connection of these relations to homology of the deleted product of a graph. We construct almost embeddings realizing some values of these invariants. This paper is accessible to mathematicians not specialized in the area (and to students). All the necessary definitions are recalled. We present some ideas of algebraic and geometric topology in a language accessible to non-topologists. However elementary, this paper is motivated by frontline of research; there are some conjectures and an open problem.