Topological interaction that arises in interlinked polymeric rings such as DNA catenanes is considered. More specifically, the free energy for a pair of linked random walk rings is derived where the distance R between two segments, each of which is part of a different ring, is kept constant. Topology conservation is imposed by the Gauss invariant. A previous approach (Otto and Vilgis 1998 Phys. Rev. Lett. 80 881) to the problem is refined in several ways. It is confirmed that asymptotically, i.e. for large R ≫ R_G where R_G is the average size of a single random walk ring, the effective topological interaction (free energy) scales ∝R⁴.