An orientable map is often presented as a realization of a finite connected graph G in an orientable surface so that the complementary domains of G, the “faces” of the map are topological open discs. This is not the definition to be used in the paper. But let us contemplate it for a while.
On each edge of G we can recognize two opposite directed edges, or “darts”. Let θ be the permutation of the dart-set S that interchanges each dart with its opposite. The darts radiating from a vertex v occur in a definite cyclic order, fixed by a chosen positive sense of rotation on the surface. The cyclic orders at the various vertices are the cycles of a permutation P of S. The choice of P rather than P –l, which corresponds to the other sense of rotation, makes the map “oriented”.