  Saddle-point van Hove singularities in the topological surface states are
interesting because they can provide a new pathway for accessing exotic
correlated phenomena in topological materials. Here, based on first-principles
calculations combined with a $\mathbf {k \cdot p}$ model Hamiltonian analysis,
we show that the layered platinum mineral jacutingaite (Pt$_2$HgSe$_3$)
harbours saddle-like topological surface states with associated van Hove
singularities. Pt$_2$HgSe$_3$ is shown to host two distinct types of nodal
lines without spin-orbit coupling (SOC) which are protected by combined
inversion ($I$) and time-reversal ($T$) symmetries. Switching on the SOC gaps
out the nodal lines and drives the system into a topological insulator state
with nonzero weak topological invariant $Z_2=(0;001)$ and mirror Chern number
$n_M=2$. Surface states on the naturally cleaved (001) surface are found to be
nontrivial with a unique saddle-like energy dispersion with type II van Hove
singularities. We also discuss how modulating the crystal structure can drive
Pt$_2$HgSe$_3$ into a Dirac semimetal state with a pair of Dirac points. Our
results indicate that Pt$_2$HgSe$_3$ is an ideal candidate material for
exploring the properties of topological insulators with saddle-like surface
states.
