abstract: One of the important results in topological quantum information is the theorem of Freedman, Larsen, and Wang that Jones braid representations are dense at non-lattice roots of unity. I will discuss a new proof of this result that does not use the classification of finite simple groups, together with related results away from roots of unity. A key idea is that the Zariski denseness is more tractable than analytic denseness, and yet can be used as the main tool to establish analytic denseness.