On surfaces glued of 2n-gons

In this paper 2n-gons and surfaces obtained through identification of 2n-gon's sides in pairs (i.e. through sewing) are considered. As well-known, one can get surface of any genus and orientability through sewing but it's very uneasy to calculate by only the polygon and the way of sewing, because to do this one need to calculate the number of vertices appearing after identification; even for small n the problem is almost impossible if one want to do this directly. There are different ways to solve the task. The canonical variant of 4q-gon sewing (2q-gon sewing) giving an orientable (unorientable) surface of genus q is well-known, as the Harer-Zagier' numbers, that are the numbers of variants of sewing a 2n-gon to an orientable surface of gunes q. In this paper we offer a new way of Euler characteristic's of obtained surface calculation (and, hence, its genus) undepending on its orientability by means of three-colour graph and information about closed surfaces topological classification.