We present an algorithm that, given a prime p as input, determines whether or not the Burau representation of the 3-strand braid group modulo p is faithful. We also prove that the representation is indeed faithful when p≤ 13. Additionally, we re-pose Salter's question on the Burau representation of B₃ over finite fields Fₚ, and solve it for every p.
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Donsung Lee (2024) studied this question.
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