Cluster structures on braid varieties

A braid variety is an affine algebraic variety associated with a simple algebraic group G and a positive braid of the corresponding type. These varieties generalise open Richardson varieties and appear in the context of symplectic topology and in the study of link invariants such as HOMFLY-PT polynomials and Khovanov-Rozansky homology. In this talk, I will give a proof of the existence of cluster A-structures and cluster Poisson structures on any braid variety. When applied to Richardson varieties, this result proves a conjecture of Bernard Leclerc. I will sketch an explicit construction of cluster seeds involving the diagrammatic calculus of weaves and explain certain properties of these cluster algebras. The talk is based on joint work with Roger Casals, Eugene Gorsky, Ian Le, Linhui Shen, and José Simental.