  Inspired by recent advances in the chromosome capture techniques, a method is
proposed to study the structural organization of systems of polymers rings with
topological constraints.To this purpose, the system is divided into
compartments and a simple condition is provided in order to determine if two
compartments are in contact or not. Next, a set of contact matrices $\bar
T_{ab}$ is defined that count how many times during a simulation a compartment
$a$ was found in contact with a non-contiguous compartment $b$ in conformations
with a given energy or temperature. Similar strategies based on correlation
maps have been applied to the study of knotted polymers in the recent past. The
advantage of the present approach is that is coupled with the Wang-Landau
algorithm. Once the density of states is computed, it is possible to generate
the contact matrices at any temperature. This gives an immediate overview over
the changes of phases that polymer systems undergo. The information on the
structure of knotted polymers and links stored in the contact matrices is the
result of averaging hundred of billions of conformations and visualized by
means of colormaps. The obtained color patterns allow to identify the main
properties of the structure of the system under investigation at any
temperature. The method is applied to detect the structural rearrangements
following the phase transitions of a knotted polymer ring and a circular
polycatenane composed by four rings in a solution. It is shown that the
colormaps have a finite number of patterns that can be clearly associated with
the different phases of these systems. Colormaps also bring new knowledge, for
instance predicting the average number of tails appearing in the conformations
of the considered polymers at a given temperature.
