  The thermodynamic criticality of the AdS black holes serves as an important
structure during the thermal phase transition. This paper discusses about the
critical points and their topology during thermal phase transitions of the
Born-Infeld AdS black holes. We make such investigations using two different
topological approaches, namely, using Duan's topological current $\phi$-mapping
theory, and the off-shell free energy. Within Duan's formalism, we observe that
for a given value of the Born-Infeld parameter $b$, there exists an associated
electric charge parameter $Q$, which is highly sensitive to the topological
phase transitions. This way we examine the connections of the first-order phase
transition and the topological nature of the critical points. We find that the
topological nature has a possible breakdown in certain parametric ranges. In
effect, we determine the unconventional and the conventional phase critical
points as the creation (topologically vortex) and annihilation (topologically
anti-vortex) points (pairs). As the second approach, we call the off-shell free
energy to determine the topological classes: of which one corresponds to the
AdS-Schwarzschild black hole phases, while the other provides a possible
topological phase transition. Here we also reveal a novel phase transition
between two unstable phases, namely, the unstable small black hole and the
intermediate black holes. For a certain parametric values of the Born-Infeld
parameter and the pressure, we also study the different topological
descriptions that inevitably correspond to the AdS-Reissner-Nordstr$\Ddot{o}$m
black hole phases. As a consistency check of the critical points during the
topological phase transitions, we study the vortex/anti-vortex annihilation
thermodynamics from local as well as global thermodynamic viewpoint.
