Formal Definition
Formally, a rotation system is defined as a pair (σ,θ) where σ and θ are permutations acting on the same ground set B, θ is a fixed-point-free involution, and the group <σ,θ> generated by σ and θ acts transitively on B.
To derive a rotation system from a 2-cell embedding of a connected multigraph G on an oriented surface, let B consist of the darts (or flags, or half-edges) of G; that is, for each edge of G we form two elements of B, one for each endpoint of the edge. Even when an edge has the same vertex as both of its endpoints, we create two darts for that edge. We let θ(b) be the other dart formed from the same edge as b; this is clearly an involution with no fixed points. We let σ(b) be the dart in the clockwise position from b in the cyclic order of edges incident to the same vertex, where "clockwise" is defined by the orientation of the surface.
If a multigraph is embedded on an orientable but not oriented surface, it generally corresponds to two rotation systems, one for each of the two orientations of the surface. These two rotation systems have the same involution θ, but the permutation σ for one rotation system is the inverse of the corresponding permutation for the other rotation system.
Read more about this topic: Rotation System
Famous quotes containing the words formal and/or definition:
“The bed is now as public as the dinner table and governed by the same rules of formal confrontation.”
—Angela Carter (19401992)
“Mothers often are too easily intimidated by their childrens negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)