Jupiters bright persistent polar aurora and Earths dark polar region indicate that the planets magnetospheric topologies are very different. High-resolution global simulations show that the reconnection rate at the interface between the interplanetary and jovian magnetic fields is too slow to generate a magnetically open, Earth-like polar cap on the timescale of planetary rotation, resulting in only a small crescent-shaped region of magnetic flux interconnected with the interplanetary magnetic field. Most of the jovian polar cap is threaded by helical magnetic flux that closes within the planetary interior, extends into the outer magnetosphere and piles-up near its dawnside flank where fast differential plasma rotation pulls the field lines sunward. This unusual magnetic topology provides new insights into Jupiters distinctive auroral morphology.