I have been doing some research on Topology in Physics and so I came across this picture
Source is this link.
Now the way I understood Topology so far is that you can classify specific characteristics of matter depending on their global characteristics. These classes seem to be divided by a topological invariant, so:
If I consider the Spins of Fermions on a surface of any matter the way they are repesented on the given picture, the matter enters a new Topological Phase whenever a new Pair of Vortices appears. But (if what I wrote above is true) this new Topological Phase is defined by another topological invariant than it had beforoe. This invariant would be (from my point of view) the number of Vortices in a system.
To clarify that: The way I understood Topological Phases and Topological Phase Transition, a new number of Vortices means a new Topological Phase. But on the given Picture the "blue" Phase and the "red" phase do have the same Number of Vortices in it (4). The only difference is, that the Tight pair of vortices became two single Vortices. That difference does not make it a new number, does it? So why is this a Topological Phase Transition?