Bounds on saddle connections for flat spheres

Abstract We consider a flat metric with conical singularities on the sphere. Under the assumption that no partial sum of angle defects is equal to , we draw on the geometry of immersed disks to obtain an explicit upper bound on the number of saddle connections with at most 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.