Reconfigurable non-Abelian braiding of nematic bits.

Non-Abelian braiding offers a route to information processing that is robust against local perturbations, yet its programmable realization in real space remains challenging. Here we present a room-temperature soft-matter platform for reconfigurable non-Abelian braiding based on the light-driven transformations of disclination lines in a nematic liquid crystal. By photonically manipulating the entangled colloids, we weave the lines into chiral double-helix entanglements and encode their topological states as nematic bits. We implement a complete set of braid operations and demonstrate their non-commutativity in three-line networks, a defining signature of non-Abelian behaviour. Repositioning colloidal gates enables in situ reprogramming, whereas the method extends to multiline architectures. Building on the scalability of this approach, we establish a predictive inverse-design framework that algebraically compiles target topological transformations into prescribed spatial routing and layer-by-layer phase corrections. These results establish a programmable classical platform for robust topological transformations and connect soft-matter physics with topological information processing.