  Two-dimensional fullerene networks have been synthesized in several forms
[Hou et al., Nature 606, 507 (2022)], and it is unknown which monolayer form is
stable at ambient condition. Using first principles calculations, I show that
the believed stability of the quasi-tetragonal phases is challenged by
mechanical, dynamic or thermodynamic stability. For all temperatures, the
quasi-hexagonal phase is thermodynamically least stable. However, the
relatively high dynamic and mechanical stabilities suggest that the
quasi-hexagonal phase is intrinsically stronger than the other phases under
various strains. The origin of the high stability and strength of the
quasi-hexagonal phase can be attributed to the strong covalent C$-$C bonds that
strongly hold the linked C$_{60}$ clusters together, enabling the closely
packed hexagonal network. These results rationalize the experimental
observations that so far only the quasi-hexagonal phase has been exfoliated
experimentally as monolayers.
