Weave-realizability for D–type

We study exact Lagrangian fillings of Legendrian links of Dₙ-type in the standard contact 3-sphere. The main result is the existence of a Lagrangian filling, represented by a weave, such that any algebraic quiver mutation of the associated intersection quiver can be realized as a geometric weave mutation. The method of proof is via Legendrian weave calculus and a construction of appropriate 1-cycles whose geometric intersections realize the required algebraic intersection numbers. In particular, we show that in D-type, each cluster chart of the moduli of microlocal rank-1 sheaves is induced by at least one embedded exact Lagrangian filling. Hence, the Legendrian links of Dₙ-type have at least as many Hamiltonian isotopy classes of Lagrangian fillings as cluster seeds in the Dₙ-type cluster algebra, and their geometric exchange graph for Lagrangian disk surgeries contains the cluster exchange graph of Dₙ-type.