Denseness and Zariski denseness of Jones braid representations

Abstract

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 obtain the same result when $t$ is a nonlattice root of unity, other than one initial case when $t$ has order 10. We also compute the real Zariski closure of these representations (meaning, the closure in Zariski closure of the real Weil restriction). When such a representation is indiscrete in the analytic topology, then its analytic closure is the same as its real Zariski closure.