  Superconducting vortices are promising traps to confine non-Abelian Majorana
quasi-particles. It has been widely believed that bulk-state topology, of
either normal-state or superconducting ground-state wavefunctions, is crucial
for enabling Majorana zero modes in solid-state systems. This common belief has
shaped two major search directions for Majorana modes, in either intrinsic
topological superconductors or trivially superconducting topological materials.
Here we show that Majorana-carrying superconducting vortex is not exclusive to
bulk-state topology, but can arise from topologically trivial quantum materials
as well. We predict that the trivial bands in superconducting HgTe-class
materials are responsible for inducing anomalous vortex topological physics
that goes beyond any existing theoretical paradigms. A feasible scheme of
strain-controlled Majorana engineering and experimental signatures for vortex
Majorana modes are also discussed. Our work provides new guidelines for
vortex-based Majorana search in general superconductors.
