  The tennis racket effect is a geometric phenomenon which occurs in a free
rotation of a three-dimensional rigid body. In a complex phase space, we show
that this effect originates from a pole of a Riemann surface and can be viewed
as a result of the Picard-Lefschetz formula. We prove that a perfect twist of
the racket is achieved in the limit of an ideal asymmetric object. We give
upper and lower bounds to the twist defect for any rigid body, which reveals
the robustness of the effect. A similar approach describes the Dzhanibekov
effect in which a wing nut, spinning around its central axis, suddenly makes a
half-turn flip around a perpendicular axis and the Monster flip, an almost
impossible skate board trick.
