I am following this paper on Grassmannians and Cluster Structures. I drew the following diagram for $Gr(2,6)$:
However, it doesn't satisfy the property that each alternating region is labelled by a $2$-subset, indicating that I have violated some rule while drawing the diagram. I cannot spot my mistake.
For reference, I am listing the rules here:
- The strands have no self-intersections.
- There are finitely many intersections and they are transversal of multiplicity $2$.
- Crossings alternate (following any strand, the strands crossing it alternate between crossing from the left and crossing from the right).
- There are no “unoriented lenses”: if two strands cross, they form an oriented disk.
I think I got it: the meeting of in-paths and out-paths at boundary vertices are also crossings—the diagram above therefore violates the condition about alternating crossings.
Even if you are using the alternate definition of a Postnikov diagram (as in Marsh's Lecture Notes on Cluster Algebras) where there are two vertices $i$ and $i'$ corresponding to each index, one for the in-path and one for the out-path, you can assume the boundary itself to have a clockwise orientation to 'complete' your boundary oriented cells.