  We discuss the topological invariant in the (2+1)-dimensional quench dynamics
of a two-dimensional two-band Chern insulator starting from a topological
initial state (i.e., with a nonzero Chern number $c_i$), evolved by a
post-quench Hamiltonian (with Chern number $c_f$). In contrast to the process
with $c_i=0$ studied in previous works, this process cannot be characterized by
the Hopf invariant that is described by the sphere homotopy group
$\pi_3(S^2)=\mathbb{Z}$. It is possible, however, to calculate a variant of the
Chern-Simons integral with a complementary part to cancel the Chern number of
the initial spin configuration, which at the same time does not affect the
(2+1)-dimensional topology. We show that the modified Chern-Simons integral
gives rise to a topological invariant of this quench process, i.e., the linking
invariant in the $\mathbb{Z}_{2c_i}$ class: $\nu = (c_f - c_i) \mod (2c_i)$. We
give concrete examples to illustrate this result and also show the detailed
deduction to get this linking invariant.
