Molecular topology of polymers plays a key role in determining their physical properties. We studied herein the topological effects on the static and dynamic properties of a 2D catenated network of DNA rings called a kinetoplast. Restriction enzymes that cleave DNA at sequence-specific sites are used to selectively cut and remove rings from the network and hence tune the molecular topology while maintaining overall structural integrity. We find that topology has minimal effects over the spatial extension of the 2D network; however, it significantly affects the relaxation behavior. The shape fluctuations of the network are governed by two distinct characteristic time scales attributed to the thermal fluctuations and confinement of the network. The relationship between the time constant of thermal relaxation and the amplitude of anisotropy fluctuations yields a universal scaling. Interestingly, this scaling is independent of the detailed arrangements of rings and/or perforation within the catenated networks. This study provides a route to tune the elastic properties of 2D catenated DNA networks and other polymeric materials by modifying the underlying topology in a rational and highly controllable manner.