  We have investigated several classes of strong spin-orbit chalcogenides
related to the (Pb,Sn)Te series studied in connection with the Dirac fermion
physics in the 1980s. Our first-principle theoretical calculations suggest that
ternary chalcogenides TlBiX$_2$ and TlSbX$_2$ (X=Te, Se, S) series harbor small
bandgap topological insulators with single Dirac cone on some selective
surfaces whereas the isostructural and isoelectronic silver-based AgBiX$_2$ and
AgSbX$_2$ (X=Te, Se, S) series do not. We find that several Tl-compounds are in
the vicinity of a topological critical point. We identify the precise surface
termination that realizes the single Dirac cone. The single-Dirac-cone surface
is shown to be correlated with a termination that minimizes dangling bonding
effects favorable for ARPES experiments. Our results further suggest that this
class of topological semi-metals may harbor odd-parity topological
superconductors similar to the possibility proposed for CuxBi2Se3 (T_c ~ 4K).
