Augmentations and immersed Lagrangian fillings

Abstract

For a Legendrian link $\Lambda \subset J^{1}M$ with $M=R$ or $S^1$, immersed exact Lagrangian fillings $L\subset \text{Symp}\left(J^{1}M\right)\cong T^{\ast }(R_{>0}\times M)$ of $\Lambda$ can be lifted to conical Legendrian fillings $\Sigma \subset J^1(R_{>0}\times M)$ of $\Lambda$. When $\Sigma$ 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 $\alpha :A\left(\Sigma \right)\to Z/2$ of the LCH algebra of $\Sigma$, there is an induced augmentation ${\epsilon }_{(\Sigma ,\alpha )}:A\left(\Lambda \right)\to Z/2$. With $\Sigma$ fixed, the set of homotopy classes of all such induced augmentations, $I_{\Sigma }\subset \mathrm{Aug}\left(\Lambda \right)/\sim$, is a Legendrian isotopy invariant of $\Sigma$. We establish methods to compute $I_{\Sigma }$ 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 $n⩾1$, we give examples of Legendrian torus knots with $2n$ distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when $\rho \ne 1$ and $\Lambda \subset J^{1}R$, every $\rho$-graded augmentation of $\Lambda$ 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 $\rho$-graded augmented Legendrian cobordism.