Fig. 3

(L) \({\tilde{\beta }}_2 > 0\) for these six (of 218 total) digraphs on four vertices. In each case it turns out that \({\tilde{\beta }}_p = \delta _{p,2}\). As a “suspension” of the 2-cycle, the digraph in the upper left can be thought of as an analogue of a homology 2-sphere obtained by gluing two cones along a common equator. (R) \({\tilde{\beta }}_3 > 0\) for these five digraphs on four vertices. In each case it turns out that \({\tilde{\beta }}_p = \delta _{p,3}\). Continuing the geometrical-topological analogy from before, the upper middle digraph in the right panel is akin to that in the upper left of the left panel with its “poles” glued together by a circular path in another dimension, thus giving rise to homology in dimension 3