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The presence of stable topological defects in a two-dimensional (\textit{d} =2) liquid crystal model allowing molecular reorientations in three dimensions(\textit{n} = 3) was largely believed to induce defect-mediatedBerzenskii-Kosterlitz-Thouless (BKT) type transition to a low temperature phasewith quasi long-range order. However, earlier Monte Carlo (MC) simulationscould not establish certain essential signatures of the transition, suggestingfurther investigations. We study this model by computing its equilibriumproperties through MC simulations, based on the determination of the density ofstates of the system. Our results show that, on cooling, the high temperaturedisordered phase deviates from its initial progression towards the topologicaltransition, crossing over to a new fixed point, condensing into a nematic phasewith exponential correlations of its director fluctuations. The thermallyinduced topological kinetic processes continue, however limited to the lengthscales set by the nematic director fluctuations, and lead to a secondtopological transition at a lower temperature. We argue that in the (\textit{d}= 2, \textit{n} = 3) system with a biquadratic Hamiltonian, the presence ofadditional molecular degree of freedom and local $Z_{2}$ symmetry associatedwith lattice sites, together promote the onset of an additional relevantscaling field at matching length scales in the high temperature region, leadingto a crossover.

Two phase transitions in the two-dimensional nematic 3-vector model with no quasi long-range order: Monte Carlo simulation of the density of states