Abstract

The theory of cluster algebras and the study of Legendrian knots, coming from the worlds of algebra and geometry respectively, are currently engaged in robust and fruitful dialogue. In this talk, we investigate positive-braid Legendrian links via a Floer-theoretic approach and prove that the coordinate ring of their augmentation varieties are cluster algebras. Using the exact Lagrangian cobordisms of Legendrian links developed by Ekholm-Honda-Kalman, we prove that the admissible Lagrangian fillings of positive-braid Legendrian links correspond to cluster seeds of their augmentation varieties. We solve the infinite-filling problem for positive-braid Legendrian links; i.e., whenever a positive-braid Legendrian link is not of type ADE , it admits infinitely many exact Lagrangian fillings. This talk is based on joint work with Honghao Gao and Daping Weng.