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

Casals, Roger; Zaslow, Eric

doi:10.2140/gt.2022.26.3589

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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

Abstract

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 $(J^{1} 𝕊^{2}, ξ_{st})$ , a new construction of infinitely many exact Lagrangian fillings for Legendrian links in $(𝕊^{3}, ξ_{st})$ , and performing $𝔽_{q}$ –rational point counts that distinguish Legendrian surfaces in $(ℝ^{5}, ξ_{st})$ . 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.

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Keywords

Legendrian, Lagrangian fillings, weaves, cluster structures, microlocal sheaves

Mathematical Subject Classification

Primary: 53D35

Secondary: 57R17

References

Publication

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

Authors

Roger Casals
Department of Mathematics University of California, Davis Davis, CA United States

Eric Zaslow
Department of Mathematics Northwestern University Evanston, IL United States

Volume 26, issue 8 (2022)