  The thermodynamics and topology of mean-field models with 2+k body
interaction terms (generalizing XY model) are derived. Focusing on two
particular cases (2+4 and 2+6 body interaction terms), a comparison between
thermodynamic (phase transition energy, thermodynamically forbidden energy
regions) and topological (singularity and curvature of saddle entropy)
properties is performed. We find that i) a topological change is present at the
phase transition energy; however, ii) only one topological change occurs, also
for those models exhibiting two phase transitions; iii) the order of a phase
transition is not completely signaled by the curvature of topological
quantities.
