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arXiv · 2101.10306

Weave Realizability for D-type

James Hughes
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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.

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James Hughes. 2021-01-25. Weave Realizability for D-type. https://doi.org/10.2140/agt.2023.23.2735

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