  Two ring polymers close to each other in space may be either in a segregated
phase if there is a strong repulsion between monomers in the polymers, or
intermingle in a mixed phase if there is a strong attractive force between the
monomers. These phases are separated by a critical point which has a
$\theta$-point character. The metric and topological properties of the ring
polymers depend on the phase, and may change abruptly at the critical point. In
this paper we examine the thermodynamics and linking of two ring polymers close
in space in both the segregated and mixed phases using a cubic lattice model of
two polygons interacting with each other. Our results show that the probability
of linking is low in the segregated phase, but that it increases through the
critical point as the model is taken into the mixed phase. We also examine the
metric and thermodynamic properties of the model, with focus on how the
averaged measures of topological complexity are related to these properties.
