We investigate the hydrodynamic role of stopping vortices in medusan propulsion through a systematic numerical study of geometrically perturbed jellyfish bells. Using fully coupled, three-dimensional, fluid-structure interaction simulations, we introduce a controlled topological modification by perforating the bell apex. This disrupts the confinement of the subumbrellar cavity and alters the dynamics of vortex formation. By varying the aperture size, we isolate the effect of structural integrity on the generation, organization, and interaction of starting and stopping vortices. Our results demonstrate that even minor perforations result in a significant decline in swimming performance, ultimately leading to negligible or even reversed net propulsion with sufficiently large apertures. Although the fundamental vortex ring structures remain, their strength, spatial distribution, and mutual alignment change significantly. Notably, stopping vortices shift toward the symmetry axis and exhibit reduced coherence. This impairs their interaction with starting vortices and suppresses the virtual-wall effect. Concurrently, an additional apical vortex emerges and grows stronger with increasing hole size, redistributing circulation away from the canonical propulsive structures. These modifications substantially reduce thrust generation and increase the cost of transport despite minor changes in bell kinematics. Our findings demonstrate that confining the subumbrellar cavity is critical for sustaining the hydrodynamic mechanisms that underlie efficient propulsion. More broadly, we highlight how targeted morphological perturbations can reveal the fluid-mechanical constraints that shape biological form and function.