Weave-realizability for D–type

Hughes, James

doi:10.2140/agt.2023.23.2735

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Weave-realizability for $D$–type

James Hughes

Algebraic & Geometric Topology 23 (2023) 2735–2776

DOI: 10.2140/agt.2023.23.2735

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.

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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Volume 23, issue 6 (2023)