Fundamentalpolygon

In mathematics, each closed in the topological sense surface can be generated by identifying the sides of a polygon with an even number of pages in pairs. This polygon is called fundamental polygon.

These polygons can be described by a character string that allocates each side a symbol. Sites that are identified with each other thereby obtained the same symbol. An additional exponent 1 or -1 indicates the orientation of the page.

Canonical form for compact surfaces ( without boundary )

In accordance with the classification set can be divided into three areas equivalence classes. Each of these classes can be assigned to a canonical form of the fundamental polygons:

• A sphere
• An orientable surface of genus
• A non-orientable surface of genus

Canonical form for compact surfaces with boundary

Surfaces with boundary differ from those without in that they additionally have a certain number of boundary components. The canonical form is obtained by expanding the fundamental polygons of unberandeten surfaces to a corresponding number of boundary components:

• A sphere with boundary components
• An orientable surface of genus with boundary components
• A non-orientable surface of genus with boundary components