Think of the Klein bottle K as the square [0,1]^2 with identifications: 1. Top edge identified to bottom edge directly: (x, 0) ~ (x, 1). 2. Left edge identified to right edge in reverse: (0, y) ~ (1, 1-y). The first identification makes the square into a cylinder. The second one identifies the two circular boundaries of the cylinder with each other by reflection about a diameter, reversing orientation. Consider any two simple closed curves in K *equivalent* if one can be continuously deformed to become the other. Puzzle: ------- Find one representative of each equivalence class of simple closed curves on K. —Dan