Augmentations and immersed Lagrangian fillings

For a Legendrian link Λ⊂ J¹M with M = R or S¹, immersed exact Lagrangian fillings L ⊂ Symp(J¹M) T^*(R>0 × M) of $Λ$ can be lifted to conical Legendrian fillings Σ⊂ J¹(R>0 × M) of $Λ$. When $Σ$ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from [30], for each augmentation α: A(Σ) → Z/2 of the LCH algebra of $Σ$, there is an induced augmentation ε(Σ,α): A(Λ) → Z/2. With $Σ$ fixed, the set of homotopy classes of all such induced augmentations, I_Σ⊂ Aug(Λ)/~, is a Legendrian isotopy invariant of $Σ$. We establish methods to compute I_Σ based on the correspondence between Morse complex families and augmentations. This includes developing a functoriality for the cellular DGA from [31] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary n ≥ 1, we give examples of Legendrian torus knots with $2n$ distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when ρ≠ 1 and Λ⊂ J¹R every $ρ$-graded augmentation of $Λ$ 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 $ρ$-graded augmented Legendrian cobordism.