For a Legendrian link $\text{◂⊂▸}\mathrm{\Lambda }\subset J^{1}M$ with $M=\mathbb{R}$ or $S^1$, immersed exact Lagrangian fillings of $\mathrm{\Lambda }$ can be lifted to conical Legendrian fillings of $\mathrm{\Lambda }$. When $\mathrm{\Sigma }$ 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 $\alpha :\text{◂→▸}\mathcal{A}(\mathrm{\Sigma })\to \mathbb{Z}/2$ of the LCH algebra of $\mathrm{\Sigma }$, there is an induced augmentation $\text{◂◽.▸}{\epsilon }_{(\mathrm{\Sigma },\alpha )}:\text{◂→▸}\mathcal{A}(\mathrm{\Lambda })\to \mathbb{Z}/2$. With $\mathrm{\Sigma }$ fixed, the set of homotopy classes of all such induced augmentations, $\text{◂⊂⋯▸}{I}_{\mathrm{\Sigma }}\subset \mathrm{Aug}(\mathrm{\Lambda })/\sim$, is a Legendrian isotopy invariant of $\mathrm{\Sigma }$. We establish methods to compute $I_{\mathrm{\Sigma }}$ 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 $\text{◂⊂▸}\mathrm{\Lambda }\subset J^1\mathbb{R}$, every $\rho$-graded augmentation of $\mathrm{\Lambda }$ 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.

中文翻译：

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增强和沉浸式拉格朗日填充

对于 Legendrian 链接$\text{◂⊂▸}\mathrm{\Lambda }\subset {杰}^{\mathrm{1个}}米$和$米=\mathbb{R}$或者${\mathrm{小号}}^{\mathrm{1个}}$, 浸入精确的拉格朗日填充的$\mathrm{\Lambda }$可以提升到圆锥形 Legendrian 馅料的$\mathrm{\Lambda }$. 什么时候$\mathrm{\Sigma }$是嵌入的，使用来自 Pan 和 Rutherford [J. 辛几何。19（2021），没有。3, 635–722], 每次增强$\alpha :\text{◂→▸}\mathcal{A}(\mathrm{\Sigma })\to \mathbb{Z}/\mathrm{2个}$LCH代数的$\mathrm{\Sigma }$, 存在诱导增强$\text{◂◽.▸}{\epsilon }_{(\mathrm{\Sigma },\alpha )}:\text{◂→▸}\mathcal{A}(\mathrm{\Lambda })\to \mathbb{Z}/\mathrm{2个}$. 和$\mathrm{\Sigma }$固定的，所有此类诱导增强的同伦类集合，$\text{◂⊂⋯▸}{我}_{\mathrm{\Sigma }}\subset \mathrm{八月}(\mathrm{\Lambda })/~$, 是勒让德同位素不变量$\mathrm{\Sigma }$. 我们建立计算方法${我}_{\mathrm{\Sigma }}$基于 MCF 和增强之间的对应关系。这包括从 Rutherford 和 Sullivan [Adv. 数学。374 (2020), 107348, 71 pp.] 关于 Legendrian 共线，并证明其等价于 LCH 的函子性。对于任意$n⩾\mathrm{1个}$，我们给出了 Legendrian 圆环结的例子$\mathrm{2个}n$独特的圆锥形 Legendrian 填充物以其诱发的增强集而著称。我们证明当$\rho \ne \mathrm{1个}$和$\text{◂⊂▸}\mathrm{\Lambda }\subset {杰}^{\mathrm{1个}}\mathbb{R}$,每个 $\rho$- 分级增强$\mathrm{\Lambda }$可以通过浸入式拉格朗日填充以这种方式诱导。或者，这被视为对适当概念的并列类的计算$\rho$- 分级增强 Legendrian cobordism。