Using various tools from representation theory and group theory, but without
using hard classification theorems such as the classification of finite simple
groups, we show that the Jones representations of braid groups are dense in the
complex Zariski topology when the parameter $t$ is not a root of unity. As
first established by Freedman, Larsen, and Wang, we the same result when t is a
non-lattice root of unity, other than one initial case when t has order 10. We
also compute the real Zariski closure of these representations.