  In this paper, we adopt an extended relativized quark model to investigate
the triply-heavy tetrquarks systematically. The mass spectra are obtained by
solving the four-body relativized Hamiltonian including the Coulomb potential,
confining potential, spin-spin interactions, and relativistic corrections. We
find that all of the triply-heavy tetraquarks lie above the corresponding
meson-meson thresholds, and thus no stable one exists. In particular, besides
the spin-spin interactions, the Coulomb and confining potentials also
contribute to the mass splittings in the $cb\bar c \bar q$ and $cb\bar b \bar
q$ systems. Moreover, the whole mass spectra for triply heavy tetraquarks show
quite similar patterns, which preserve the light flavor SU(3) symmetry and
heavy quark symmetry well. Through the fall-apart mechanism, the triply-heavy
tetraquarks may easily decay into the heavy quarkonium plus heavy-light mesons,
which are good candidates for investigation in future experiments.
