  We study the prerequisites for realizing superconductivity in doped
triangular-lattice Mott insulators by considering three distinct parent spin
backgrounds, i.e., $120^{\circ}$ antiferromagnets, quantum spin liquid, and
stripy antiferromagnets, and all possible sign combinations $(\tau_1, \tau_2)$
of nearest-neighbor hopping and next-nearest-neighbor hopping $(t_1, t_2)$.
Based on density-matrix renormalization group calculations, we find that, with
finite $t_2$ and specific sign combinations $(\tau_1, \tau_2)$, the
quasi-long-range superconductivity order can always be achieved, regardless of
the nature of the parent spin backgrounds. Besides specific hopping signs
$(\tau_1, \tau_2)$, these superconductivity phases in triangular lattices are
commonly characterized by short-ranged spin correlations and two charges per
stripe. In the robust superconductivity phase realized at larger $t_2/t_1$,
flipping the signs $\tau_2$ and $\tau_1$ gives rise to the stripe phase without
strong pairing and pseudogap-like phase without Cooper-pair phase coherence,
respectively. Interestingly, the roles of the two hopping signs are switched at
smaller $t_2/t_1$. Moreover, different sign combinations $(\tau_1, \tau_2)$
would stabilize distinct phases including superconductivity, charge density
waves, spin density waves, and pseudogap-like phases accordingly. Our findings
suggest the important role of charge kinetic energy in realizing
superconductivity in doped triangular-lattice Mott insulators.
