Recovering The Embedding From The Rotation System
To recover a multigraph from a rotation system, we form a vertex for each orbit of σ, and an edge for each orbit of θ. A vertex is incident with an edge if these two orbits have a nonempty intersection. Thus, the number of incidences per vertex is the size of the orbit, and the number of incidences per edge is exactly two. If a rotation system is derived from a 2-cell embedding of a connected multigraph G, the graph derived from the rotation system is isomorphic to G.
To embed the graph derived from a rotation system onto a surface, form a disk for each orbit of σθ, and glue two disks together along an edge e whenever the two darts corresponding to e belong to the two orbits corresponding to these disks. The result is a 2-cell embedding of the derived multigraph, the two-cells of which are the disks corresponding to the orbits of σθ. The surface of this embedding can be oriented in such a way that the clockwise ordering of the edges around each vertex is the same as the clockwise ordering given by σ.
Read more about this topic: Rotation System
Famous quotes containing the words rotation and/or system:
“The lazy manage to keep up with the earths rotation just as well as the industrious.”
—Mason Cooley (b. 1927)
“Science is a system of statements based on direct experience, and controlled by experimental verification. Verification in science is not, however, of single statements but of the entire system or a sub-system of such statements.”
—Rudolf Carnap (18911970)