n. 1. a measure of the connectedness of a closed surface, equal to 1-K/4π, for K the integral Gaussian curvature. See also Euler's formula. 2. (of an algebraic plane curve) the difference between the maximum number of double points a curve of the given degree may possess and the actual number of the given curve. 3. (for a topological surface) a pair (p, q), where p is the number of handles and q is the number of cross-caps of the surface. 4. a class of non-equivalent primitive binary quadratic forms with given discriminant, each form representing the same integers. 5. the least natural number m such that an entire function has a Weierstrass product expansion where g is entire, and n ranges over the natural numbers. If no such expression exists the genus is infinite. Compare order.