a. Diagonal Black and White holes in the two dimensional ω-plane. Only two direct paths can link points a and b which are continuously deformable into one another without crossing either hole. There are two additional monotonic switchback paths which are not drawn. Equivalence classes of paths define the fundamental dihomotopy groupoid. b. Cross-diagonal Black and White holes as in 3a. Three direct equivalence classes of continuously deformable paths can link a and b. Thus the two spaces are topologically distinct, having different dihomotopy groupoids. Here monotonic switchbacks are not possible, although relaxation of that condition can lead to 'backwards' switchbacks and intermediate loopings.