For n=3, the Burau representation is faithful.
For n \geq 5, the Burau representation is unfaithful.
As I know, no significant results are since Bigelow's work on 2001. The n=4 case is still open.
It is open, but you can see the flowing paper
https://arxiv.org/pdf/1705.02641.pdf
it may be interesting for you
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