The statistical mechanics of [2]catenanes in a solution with constrained number of maxima and minima along a special direction (the height) is discussed. The interest in this system comes from the fact that, in the homopolymer case, it has analogies with self-dual anyon field theory models and has conformations that minimize the static energy and bear particular symmetries and properties. In the first part of this work we provide a procedure for deriving the equations of motion in the case of replica field theories in the limit of zero replicas. We compute also the partition function of the [2]catenane at the lowest order in the frame of the so-called background field method. In the second part the statistical mechanics of the system is investigated using numerical simulations based on the Wang-Landau Monte Carlo method. At equilibrium, independently of the temperature, it turns out that the conformations of the system are elongated in the height directions. The two rings composing the [2]catenanes have approximately the same heights and are aligned. The thermodynamic properties of the system are discussed and the results coming from the field theoretical approach are compared with those of the numerical simulations.

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