Low-dimensional unitary representations of $B_3$

We characterize all simple unitarizable representations of the braid group $B_3$on complex vector spaces of dimension $d \leq 5$. In particular, we prove that if $\sigma_1$ and $\sigma_2$ denote the two generating twists of $B_3$, then a simple representation $\rho:B_3 \to \operatorname{GL} (V)$ (for $\dim V \leq 5$) is unitarizable if and only if the eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_d$ of $\rho(\sigma_1)$ are distinct, satisfy $\vert\lambda_i\vert=1$ and $\mu^{(d)}_{1i} > 0$ for $2 \leq i \leq d$, where the $\mu^{(d)}_{1i}$ are functions of the eigenvalues, explicitly described in this paper.