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 |${\mathcal {B}}_{3}$|⁠. 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 |$t\in {\mathbb {C}}^{*}$| outside the annulus |$3-2\sqrt 2 \leq |t| \leq 3+2\sqrt 2.$| [ABSTRACT FROM AUTHOR]