We analyze finite orbits of the natural braid group action on the character variety of the $n$ times punctured sphere. Building on recent results relating middle convolution and finite complex reflection groups, our work implements Katz's middle convolution to explicitly classify finite orbits in the $SL_2(\C)$-character variety of the punctured sphere. We provide theoretical results on the existence of finite orbits arising from the imprimitive finite complex reflection groups and formulas for constructing such examples when they exist. In the primitive finite complex reflection groups, we perform an exhaustive search and provide computational results. Our contributions also include Magma computer code for middle convolution and for computing the orbit under this action when it is known to be finite.

60 pages, formatted for journal

• 13-04 - Software, source code, etc. for problems pertaining to commutative algebra
• 14H30 - Coverings of curves, fundamental group
• 37-04 - Software, source code, etc. for problems pertaining to dynamical systems and ergodic theory
• 37C05 - Dynamical systems involving smooth mappings and diffeomorphisms