On intersection of two embedded spheres in 3-space

We study intersection of two polyhedral spheres without self-intersections in 3-space. We find necessary and sufficient conditions on sequences x = x_1,x_2,…,x_n, y = y_1,y_2,…,y_n of positive integers, for existence of 2-dimensional polyhedra f,g in R^3 homeomorphic to the sphere and such that

• f-g has n connected components, of which the i-th one has x_i neighbors in f and
• g-f has n connected components, of which the i-th one has y_i neighbors in g. Analogously we study intersection of three polyhedral spheres without self-intersections in 3-space. Russian version is accessible to high-school teachers and students interested in mathematics.