Weave Realizability for D-type

Abstract

We study exact Lagrangian fillings of Legendrian links of $D_n$-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 $D_n$-type have at least as many Hamiltonian isotopy classes of Lagrangian fillings as cluster seeds in the $D_n$-type cluster algebra, and their geometric exchange graph for Lagrangian disk surgeries contains the cluster exchange graph of $D_n$-type.

arXiv comment30 pages, typos corrected, figures updated

References in corpus (11)

• Legendrian knots and constructible sheaves
• Cluster varieties from Legendrian knots
• Lagrangian concordance of Legendrian knots
• Introduction to Cluster Algebras. Chapters 1-3
• Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems
• Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
• Augmentations, Fillings, and Clusters
• Satellites of Legendrian knots and representations of the Chekanov-Eliashberg algebra
• Exact Lagrangian Fillings of Legendrian $(2,n)$ torus links
• Infinitely many Lagrangian fillings
• Lagrangian fillings for Legendrian links of affine type

Cited by in corpus (3)

• Lagrangian Fillings in A-type and their Kalman Loop Orbits
• A Lagrangian filling for every cluster seed
• Non-fillable augmentations of twist knots