Let G be a complex simple group. Let $\beta$ be a positive braid whose Demazure product is the longest Weyl group element. The braid variety X($\beta$) generalizes many well known varieties, including positroid cells, open Richardson varieties, and double Bott-Samelson cells. We provide a concrete construction of the cluster structure on X($\beta$), using the weaves of Casals and Zaslow. We show that the coordinate ring of X($\beta$) is a cluster algebra, which confirms a conjecture of Leclerc as special cases. As an application, we show that X($\beta$) admits a natural Poisson structure and can be further quantized. If
time permits, I will explain several of its applications on representation theory and knot theory,
including its connections with the Kazhdan-Lusztig R-polynomials and a geometric interpretation of the
Khovanov-Rozansky homology (following the work of Lam-Speyer and Galashin-Lam). This talk is based on joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, and Jose Simental (arXiv:2207.11607).