  We reveal how the vector field links are untied under the influence of
anti-parity-time-symmetric couplings in a dissipative sublattice-symmetric
topological photonic crystal lattice. The topology of the quasi-one-dimensional
two-band system is encoded in the geometric topology of the vector fields
associated with the Bloch Hamiltonian. The linked vector fields reflect the
topology of the nontrivial phase. The topological phase transition occurs
concomitantly with the untying of the vector field link at the exceptional
points. Counterintuitively, more dissipation constructively creates a
nontrivial topology. The linking number predicts the number of topological
photonic zero modes.
