Topology of potential hypersurfaces of two, three and four dipoles interacting at long distances

The whole sets of critical points of analytical functions corresponding to the long‐range two‐body interaction between two, three and four dipole vectors set at the vertices of several polygons are determined using topology theorems. Betti numbers associated to the configuration spaces are first obtained, then stationary points are located via an analytical gradient method till the Morse inequalities are checked.