  If superconductivity is induced in the metallic surface states of topological
insulators via proximity, Majorana modes will be trapped on the vortex cores.
The same effects hold for doped topological insulators which become bulk s-wave
superconductors as long as the doping does not exceed a critical values $
\mu^{\pm}_c.$ It is this critical chemical potential at which the material
forgets it arose from a band-inverted topological insulator; it loses its
topological \emph{imprint.} For the most common classes of topological
insulators, which can be modeled with a minimal 4-band Dirac model the values
of $\mu^{\pm}_c$ can be easily calculated, but for materials with more
complicated electronic structures such as HgTe or ScPtBi the result is unknown.
We show that due to the hybridization with an additional Kramers' pair of
topologically trivial bands the topological imprint of HgTe-like electronic
structures (which includes the ternary Heusler compounds) can be widely
extended for p-doping. As a practical consequence we consider the effects of
such hybridization on the range of doping over which Majorana modes will be
bound to vortices in superconducting topological insulators and show that the
range is strongly extended for p-doping, and reduced for n-doping. This leaves
open the possibility that other topological phenomena may be stabilized over a
wider range of doping.
