Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications

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 multi-colored planar combinatorics. Second, we present an algebraic-geometric 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 subfactor of a 3-dimensional Lagrangian concordance monoid for a Legendrian surface in the 1-jet space of the two-sphere, a new construction of infinitely many exact Lagrangian fillings for Legendrian links in the standard contact three-sphere, and performing rational point counts over finite fields that distinguish Legendrian surfaces in the standard five-dimensional Darboux chart. In addition, the manuscript develops 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-Williamson's Soergel Calculus.

arXiv comment116 Pages, 106 Figures. A published version of this manuscript will appear in Geometry & Topology

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• Augmentations, Fillings, and Clusters
• Domain wall formation and magnon localization in twisted chromium trihalides
• Weave Realizability for D-type
• Lagrangian fillings for Legendrian links of affine type
• A note on infinite number of exact Lagrangian fillings for spherical spuns
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• Augmented Legendrian cobordism in $J^1{S}^1$
• Lagrangian cobordism functor in microlocal sheaf theory I
• Symplectomorphisms of some Weinstein 4-manifolds