Augmentations and immersed Lagrangian fillings

Abstract

AbstractFor a Legendrian link with or , immersed exact Lagrangian fillings of can be lifted to conical Legendrian fillings of . When is embedded, using the version of functoriality for Legendrian contact homology (LCH) from Pan and Rutherford [J. Symplectic Geom. 19 (2021), no. 3, 635–722], for each augmentation of the LCH algebra of , there is an induced augmentation . With fixed, the set of homotopy classes of all such induced augmentations, , is a Legendrian isotopy invariant of . We establish methods to compute based on the correspondence between MCFs and augmentations. This includes developing a functoriality for the cellular differential graded algebra from Rutherford and Sullivan [Adv. Math. 374 (2020), 107348, 71 pp.] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary , we give examples of Legendrian torus knots with distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when and , every ‐graded augmentation of can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of ‐graded augmented Legendrian cobordism.