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Legendrian weaves: $N$–graph calculus, flag moduli and applications

Roger Casals and Eric Zaslow

Geometry & Topology 26 (2022) 3589–3745
DOI: 10.2140/gt.2022.26.3589

We study a class of Legendrian surfaces in contact five-folds by encoding their wavefronts via planar combinatorial structures. We refer to these surfaces as Legendrian weaves, and to the combinatorial objects as N–graphs. First, we develop a diagrammatic calculus which encodes contact geometric operations on Legendrian surfaces as multicolored planar combinatorics. Second, we present an algebrogeometric characterization for the moduli space of microlocal constructible sheaves associated to these Legendrian surfaces. Then we use these N–graphs and the flag moduli description of these Legendrian invariants for several new applications to contact and symplectic topology.

Applications include showing that any finite group can be realized as a subquotient of a 3–dimensional Lagrangian concordance monoid for a Legendrian surface in (J1𝕊2,ξst ), a new construction of infinitely many exact Lagrangian fillings for Legendrian links in (𝕊3,ξst ), and performing 𝔽q–rational point counts that distinguish Legendrian surfaces in (5,ξst ). In addition, we develop the notion of Legendrian mutation, studying microlocal monodromies and their transformations. The appendix illustrates the connection between our N–graph calculus for Lagrangian cobordisms and Elias, Khovanov and Williamson’s Soergel calculus.

Legendrian, Lagrangian fillings, weaves, cluster structures, microlocal sheaves
Mathematical Subject Classification
Primary: 53D35
Secondary: 57R17
Received: 17 December 2020
Revised: 8 August 2021
Accepted: 5 September 2021
Published: 16 March 2023
Proposed: Ciprian Manolescu
Seconded: András I Stipsicz, Leonid Polterovich
Roger Casals
Department of Mathematics
University of California, Davis
Davis, CA
United States
Eric Zaslow
Department of Mathematics
Northwestern University
Evanston, IL
United States