From disc patterns in the plane to character varieties of knot groups

Motivated by an experimental study of groups generated by reflections in planar patterns of tangent circles, we describe some methods for constructing and studying representation spaces of holonomy groups of infinite volume hyperbolic $3$-manifolds that arise from unknotting tunnels of links. We include full descriptions of our computational methods, which were guided by simplicity and generality rather than by being particularly efficient in special cases. This makes them easy for non-experts to understand and implement to produce visualisations that can suggest conjectures and support algebraic calculations in the character variety. Throughout, we have tried to make the exposition clear and understandable for graduate students in geometric topology and related fields.