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

TL;DR

This paper introduces Legendrian weaves and N-graphs to encode Legendrian surfaces, develops a diagrammatic calculus, characterizes their moduli spaces algebraically, and applies these tools to solve problems in contact and symplectic topology.

Contribution

It presents a new combinatorial and algebraic framework for Legendrian surfaces, including a diagrammatic calculus and applications to topology and Lagrangian fillings.

Findings

Realized any finite group as a subfactor of a Lagrangian concordance monoid.

Constructed infinitely many exact Lagrangian fillings for Legendrian links.

Performed rational point counts distinguishing Legendrian surfaces.

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.