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Weave-realizability for $D$–type

James Hughes

Algebraic & Geometric Topology 23 (2023) 2735–2776
Abstract

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.

Keywords
Legendrian knots, exact Lagrangian fillings
Mathematical Subject Classification
Primary: 53D12
Secondary: 57K33
References
Publication
Received: 12 March 2021
Revised: 12 October 2021
Accepted: 22 March 2022
Published: 7 September 2023
Authors
James Hughes
Department of Mathematics
University of California, Davis
Davis, CA
United States

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