  The recent discovery of triply degenerate points (TDPs) in topological
materials has opened a new perspective toward the realization of novel
quasiparticles without counterparts in quantum field theory. The emergence of
such protected nodes is often attributed to spin-vector-momentum couplings.
Here we show that the interplay between spin-tensor- and spin-vector-momentum
couplings can induce three types of TDPs, classified by different monopole
charges ($\mathcal{C}=\pm 2,\pm 1,0$). A Zeeman field can lift them into Weyl
points with distinct numbers and charges. Different TDPs of the same type are
connected by intriguing Fermi arcs at surfaces, and transitions between
different types are accompanied by level crossings along high-symmetry lines.
We further propose an experimental scheme to realize such TDPs in cold-atom
optical lattices. Our results provide a framework for studying
spin-tensor-momentum coupling-induced TDPs and other exotic quasiparticles.
