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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.
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