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.