Knot as a complete invariant of the diffeomorphism of surfaces with three periodic orbits

It is known that Morse–Smale diffeomorphisms with two hyperbolic periodic orbits exist only on the sphere and they are all topologically conjugate to each other. However, if we allow three orbits to exist then the range of manifolds admitting them widens considerably. In particular, the surfaces of arbitrary genus admit such orientation-preserving diffeomorphisms. In this article we find a complete invariant for the topological conjugacy of Morse–Smale diffeomorphisms with three periodic orbits. The invariant is completely determined by the homotopy type (a pair of coprime numbers) of the torus knot which is the space of orbits of an unstable saddle separatrix in the space of orbits of the sink basin. We use the result to calculate the exact number of the topological conjugacy classes of diffeomorphisms under consideration on a given surface as well as to relate the genus of the surface to the homotopy type of the knot.