  Deploying an analytical atomistic model of the bulk band structure of the
IV-VI semiconductors we connect the spin structure of the topological surface
state to the crystal field and spin orbit coupling parameters of the bulk
material. While the Dirac-Weyl (or equivalently, Rashba) type topological
surface state is often assumed universal, we show that the physics of the
surface state is strikingly non-universal. To see this explicitly we calculate
the RKKY interaction, which may be viewed as a probe of this surface state spin
structure, finding its \emph{qualitative form} depends on the values the bulk
spin-orbit and crystal field parameters take. This opens the way to tune the
spin interaction on the surface of a IV-VI topological insulator by, for
instance varying the composition of the IV-VI ternary compounds, as well as
highlighting the importance of the connection between bulk and surface physics
in topological insulators.
