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In the creation of Hopf topological matters, the old paradigm is to conceivethe Hopf invariant first, and then display its intuitive topology throughlinks. Here we brush aside this effort and put forward a new recipe forunraveling the quenched two-dimensional (2D) two-band Chern insulators under aparallel quench protocol, which implies that the quench quantities withdifferent momentum k are parallel or antiparallel to each other. We find thatwhether the dynamical Hopf invariant exists or not, the links in (2+1)D spacealways keep their standard shape even for topological initial states, and traceout the trajectories of phase vortices. The linking number is exactly equal tothe difference between pre- and post-quench Chern numbers regardless of theconstruction of homotopy groups. We employ two concrete examples to illustratethese results, highlighting the polarity reversal at fixed points.

From orthogonal link to phase vortex in generalized dynamical Hopf insulators