I will introduce and discuss a remarkable class of algebraic varieties, called braid varieties. These include all open Richardson and positroid varieties, and are closely related to augmentation varieties for Legendrian links. The topology of braid varieties is related to various link invariants such as HOMFLY polynomial and Khovanov-Rozansky homology, while their coordinate ring has a cluster structure.

The talk is based on joint works with Roger Casals, Mikhail Gorsky, Ian
Le, Linhui Shen and Jose Simental.