  We study by Monte Carlo simulations and scaling analysis two models of pairs
of confined and dense ring polymers in two dimensions. The pair of ring
polymers are modelled by squared lattice polygons confined within a square
cavity and they are placed in relation to each other to be either unlinked or
linked in the plane. The observed rich phase diagrams of the two models reveal
several equilibrium phases separated by first order and continuous phase
boundaries whose critical nature depend on this reciprocal placements. We
estimate numerically the critical exponents associated with the phase
boundaries and with the multicritical points where first order and continuous
phase boundaries meet.
