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by jlev1 3 days ago
There was an excellent Dance Your PhD video about representations of braid groups and faithfulness in 2017: https://www.youtube.com/watch?v=MASNukczu5A

If you look closely, you can find a mention of the n=4 Burau representation in it, and the fact that it "remains an open problem" to find elements of its kernel, i.e. to determine whether it is faithful.