What is difference between annulus (cylinder) and disk in graph routing?
I know annulus is disk with hole, or I can imagine how is similar to cylinder, but my problem is, I can't understand this paragraph:
A fundamental difference between the disc and cylinder problems is that in the case of the cylinder there is more than one homotopically inequivalent route between pairs of points of $bd(\Sigma)$, even if we restrict ourselves to non-self-intersecting routes, which we can.
I can't understand why we have more than one different route between two points on annulus but one route on disk? (I think in both cases we should have one route).