Homotopy classification of closed polygonal lines

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.