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We study the problem of deciding whether or not the image of an irreduciblerepresentation of the braid group $\B_3$ of degree $\leq 5$ has finite image ifwe are only given the eigenvalues of a generator. We provide a partialalgorithm that determines when the images are finite or infinite in all butfinitely many cases, and use these results to study examples coming fromquantum groups. Our technique uses two classification theorems and thecomputational group theory package GAP.

Finite Linear Quotients of $\B_3$ of Low Dimension