We present a method for locally injective seamless parametrization of triangular mesh surfaces of arbitrary genus, with or without boundaries, given desired cone points and rational holonomy angles (multiples of 2π/ q for some positive integer q ). The basis of the method is an elegant generalization of Tutte's "spring embedding theorem" to this setting. The surface is cut to a disk and a harmonic system with appropriate rotation constraints is solved, resulting in a harmonic global parametrization (HGP) method. We show a remarkable result: that if the triangles adjacent to the cones and boundary are positively oriented, and the correct cone and turning angles are induced, then the resulting map is guaranteed to be locally injective. Guided by this result, we solve the linear system by convex optimization, imposing convexification frames on only the boundary and cone triangles, and minimizing a Laplacian energy to achieve harmonicity. We compare HGP to state-of-the-art methods and see that it is the most robust, and is significantly faster than methods with comparable robustness.