Burau representation of braid groups and $q$-rationals

Sophie Morier-Genoud; Valentin Ovsienko; Alexander Veselov

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arXiv · 2309.04240

Burau representation of braid groups and $q$-rationals

Sophie Morier-Genoud·Valentin Ovsienko·Alexander Veselov

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Abstract

We establish a link between the new theory of $q$-deformed rational numbers and the classical Burau representation of the braid group $\mathrm{B}_3$. 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 $3-2\sqrt2 \leq |t| \leq 3+2\sqrt2.$

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Sophie Morier-Genoud, Valentin Ovsienko, Alexander Veselov. 2023-09-08. Burau representation of braid groups and $q$-rationals. https://arxiv.org/abs/2309.04240

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