# Orientable Surface

## orientable surface

[‚ȯr·ē‚en·tə·bəl ′sər·fəs]
(mathematics)
A surface for which an object resting on one side of it cannot be moved continuously over it to get to the other side without going around an edge.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Orientable Surface

a surface that can be oriented. An orientable surface is the opposite of a nonorientable surface. On a nonorientable surface, for example, a Möbius band, there always exist closed curves such that the orientation of a small neighborhood of a point moving along the curve is reversed when the entire curve is traversed. The projective plane is an important example of a closed nonorientable surface.

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In particular, we obtained a homeomorphism [M.sub.5]([pi]/2) [congruent to] [[summation].sub.5], where [[summation].sub.5] denotes a connected closed orientable surface of genus 5.
For i = 0,1, 2, ..., let [g.sub.i](G) be the number of topologically distinct cellular embeddings of the graph G in the orientable surface [S.sub.i] of genus i.
A map is a connected graph embedded in a compact connected orientable surface in such a way that the regions delimited by the graph, called faces, are homeomorphic to open discs.
Let M be a 3-manifold which is constructed as a fiber bundle over a closed orientable surface with fiber a circle.
Throughout this paper, S will be an orientable surface from [R.sup.3.sub.1] and n its unit normal vector field.
Certainly, on an orientable surface, an equivalence preserve the orientation on this surface.
Henceforth, by surface we shall always mean an orientable surface, equivalently, one which is embeddable in [R.sup.3].
A map M is an embedding of a connected graph G, with possibly multi-edges or loops, into a closed, connected and orientable surface S such that all faces, i.e.
This graph is embedded in a locally orientable surface such that if we cut the graph from the surface, the remaining part consists of connected components called faces or cells, each homeomorphic to an open disk.
A unicellular map is a graph embedded on a compact orientable surface, in such a way that its complement is a topological polygon.
Bender and Canfield ([BC86]) obtained the asymptotic number of maps on a given orientable surface. Gao ([Gao93]) obtained formulas for the asymptotic number of 21-angulations on orientable surfaces, and conjectured a formula for more general families (namely maps where the degrees of the faces are restricted to lie in a given finite subset of 2N).

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