Topological transition enabled by composite symmetry-breaking paths in trefoil-knot honeycomb lattices

Topological phases are governed by lattice symmetries, yet how different symmetry-breaking paths (SBPs) affect topological transitions remains insufficiently understood. Most existing studies rely on a single SBP, and address only one bandgap, limiting independent control of multiple gaps. Here, we investigate multiple isolated Dirac points in a trefoil-knot-modified honeycomb lattice, and show that a single SBP generally inverts all relevant Dirac points simultaneously, whereas the tailored combinations of SBPs enable selective and programmable band inversion at targeted gaps. The excitation-dependent responses reveal strong modal selectivity. This capability is exploited to realize independently controllable multi-channel signal splitting, which is unattainable with a single SBP. The results enable SBPs as an effective design degree of freedom for programmable and reconfigurable topological elastic devices.