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You possibly already know the Euler formula (Equation 9.1, “Euler formula”) in three dimensions (v: vertices, e: edges, f: faces, g: genus, 2 - 2g: Euler characteristic):
The genus is a topologic invariant property of the surface. It is defined as the largest number of non-intersecting simple closed curves that can be drawn without separating it. In cases with boundaries holes count as normal faces in fulfilling the Euler formula.