We discuss the topological invariant in the (2+1)-dimensional quench dynamics of a two-dimensional twoband 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 ). This process is classified by the torus homotopy group τ 3 (S 2 ). 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 π 3 (S 2 ) = 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 Z 2c i class: ν = (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. arXiv:1904.12552v2 [cond-mat.quant-gas]