Many cluster varieties can be realized as braid varieties, including double Bruhat cells, double Bott–Samelson cells, and positroid varieties. Their cluster structures arise from type A Demazure weaves, whose Legendrian projections give Lagrangian fillings of Legendrian positive braid links. Kontsevich-Soibelman DT series and the associated cluster Donaldson–Thomas transformations provide fundamental invariants of these braid varieties. In this talk, we present a new method for computing cluster Donaldson–Thomas transformations for type A Demazure weaves using Gaiotto–Moore–Neitzke spectral networks. This talk is based on a joint project with Daping Weng.