Topological phase transitions happen when the band gap closes. It is not true that all band crossings are topological.
There are Dirac (linear) band crossings, quadratic band crossings, Dirac-like triply degenerate band crossings, double Dirac cone crossings, semi-Dirac transitions (linear in one direction and quadratic in another) etc.
Even in 1D, all the band crossings I recall look linear. In 2D, all the band crossings I recall are Dirac cones. I feel like I have been told that some quadratic dispersions can be a topological transition but I am not sure if I remember correctly.
Are all topological phase transitions in electronic bands/ photonic bands linear/ Dirac points?