Burau representation of braid groups and q-rationals

We establish a link between the new theory of q-deformed rational numbers and the classical Burau representation of the braid group B₃. We apply this link to the open problem of classification of faithful complex specializations of this representation. As a result we provide an answer to this problem in terms of the singular set of the q-rationals and prove the faithfulness of the Burau representation specialized at complex t ∈ C^∗ outside the annulus 3 − 2 √ 2 ≤ |t| ≤ 3 + 2√ 2.