  Dyonic black holes with quasitopological electromagnetism exhibit an
intriguing phase diagram with two separated first-order coexistence curves. In
this paper, we aim to uncover its influence on the black hole thermodynamical
topology. At first, we investigate the phase transition and phase diagram of
the dyonic black holes. Comparing with previous study that there is no black
hole phase transition region for a middle pressure, we find this region can
narrow or disappear by fine tuning the coupling parameter. Instead, two
first-order phase transitions can be observed. Importantly, we uncover that
such novel phase diagram shall lead to a multiple defect curve phenomenon in
black hole topology where each dyonic black hole is treated as one defect in
the thermodynamical parameter space. By examining the topology, it is shown
that there could be one, three, or five black hole states for given pressure
and temperature. For each case, the topological number is calculated. Our
results show that the topological number always takes value of +1, keeping
unchanged even when the multiple defect curves appear. Therefore, our study
provides an important ingredient on understanding the black hole
thermodynamical topology.
