Bounds on saddle connections for flat spheres

We consider a flat metric with conical singularities on the sphere. Under the assumption that no partial sum of angle defects is equal to $2\pi$, we draw on the geometry of immersed disks to obtain an explicit upper bound on the number of saddle connections with at most $k$ self-intersections. Additionally, we establish an upper bound on their lengths for a surface with a normalized area. Finally, we apply these bounds to the counting of singular trajectories in irrational polygonal billiards.