To me, it's explicitly about being topologically equivalent. I brushed up on my topology and knot theory when working on this to make sure I had it right. The post is considering two social dancers as connected components of two topological objects and the 15 different ways are all the different ways specific points of the surface (hands reduced to points that is) are able to connect.
There's more topology as well when you start bringing in other characteristics of a position, like relative distance and orientation of the dances, you can find more equivalence classes for the other types of equivalences that fall out of that.
There's more topology as well when you start bringing in other characteristics of a position, like relative distance and orientation of the dances, you can find more equivalence classes for the other types of equivalences that fall out of that.