  The topological interaction arising 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. The topology
conservation is imposed by the Gauss invariant. A previous approach (M.Otto,
T.A. Vilgis, Phys.Rev.Lett. {\bf 80}, 881 (1998)) to the problem is refined in
several ways. It is confirmed, that asymptotically, i.e. for large $R\gg R_G$
where $R_G$ is average size of single random walk ring, the effective
topological interaction (free energy) scales $\propto R^4$.
