  We present the two-loop virtual amplitudes for the production of a top-quark
pair in gluon fusion. The evaluation method is based on a numerical solution of
differential equations for master integrals in function of the quark velocity
and scattering angle starting from a boundary at high-energy. The results are
given for the renormalized infrared finite remainders on a large grid and have
recently been used in the calculation of the total cross sections at the
next-to-next-to-leading order. For convenience, we also give the known results
for the quark annihilation case on the same grid. Outside of the kinematical
range covered by the grid, we provide threshold and high-energy expansions.
  From expansions of the two-loop virtual amplitudes, we determine the
threshold behavior of the total cross sections at next-to-next-to-leading order
for the quark annihilation and gluon fusion channels including previously
unknown constant terms. In our analysis of the quark annihilation channel, we
uncover the presence of a velocity enhanced logarithm of Coulombic origin,
which was missed in a previous study.
