### Algebraic Geometry Seminar, UC Davis

Daping Weng, Michigan State University

Augmentations, Fillings, and Clusters

Organizer's time: 2020-11-18 11:00 America/Los_Angeles

Duration: 1 hour

A Legendrian link is a 1-dimensional closed manifold that is embedded in $R^3$ and satisfies certain tangent conditions. Rainbow closures of positive braids are natural examples of Legendrian links. In the study of Legendrian links, one important task is to distinguish different exact Lagrangian fillings of a Legendrian link, up to Hamiltonian isotopy, in the $R^4$ symplectization. We introduce a cluster K2 structure on the augmentation variety of the Chekanov-Eliashberg dga for the rainbow closure of any positive braid. Using the Ekholm-Honda-Kalman functor from the cobordism category of Legendrian links to the category of dga’s, we prove that a big family of fillings give rise to cluster seeds on the augmentation variety of a positive braid closure, and these cluster seeds can in turn be used to distinguish non-Hamiltonian isotopic fillings. Moreover, by relating a cyclic rotation concordance on a positive braid closure with the Donaldson-Thomas transformation on the corresponding augmentation variety, we prove that other than a family of positive braids that are associated with finite type quivers, the rainbow closure of all other positive braids admit infinitely many non-Hamiltonian isotopic fillings. This is joint work with H. Gao and L. Shen.