# Enumerating homologies of disjoint paths

I am reading this recent paper by Schrijver, in particular, section 4.2: Enumerating homologies of disjoint paths.

I did not understand how do they re-route the paths through a spanning tree and obtain the walks with the mentioned properties, so I decided to work out an example. This is what I have:

A planar graph with two pairs of terminals and want to find vertex disjoint paths between them (that is $$F=\{\{1,2\}\}$$). I have also marked a spanning tree $$T$$ (purple highlighted edges) and numbered the edges $$e_1, e_2, e_3, e_4$$ not in the tree such that if $$Q_{e_j}$$ is longer than $$Q_{e_i}$$ then $$j > i$$. (I haven't marked the direction of the edges because we are considering undirected walks)

Now I have marked two disjoint paths between the terminals. And according to the algorithm next I am supposed to eliminate $$e_2$$ from $$P_2$$ and then $$e_4$$ from $$P_1$$.

But notice that the two walks now cross each other at the top-middle vertex, but the algorithm claimed that the walks obtained will be non-crossing.

Can someone please point out my mistake. Any help would be really appreciated.

Even in this 1994 paper, they described a way to enumerate homologies but this method, as it is described, seems more difficult for me to understand.

Edit: Following is the only way, I was able to find, to re-route those paths through the tree, but as far as I understand, this is not what the paper describes.