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

We study a class of Legendrian surfaces in contact five-folds by encoding their wave-fronts 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 Leg-endrian 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 .J¹ S²;_st /, a new construction of infinitely many exact Lagrangian fillings for Leg-endrian links in .S³;_st /, and performing F_q –rational point counts that distinguish Legendrian surfaces in .R⁵;_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.