Character variety methods in 3-manifold topology

I will give a survey-ish overview of an affine algebraic set associated to a hyperbolic 3-manifold, which 3-manifold topologists tend to call the "character variety" even though it is often not a variety and may not consist of characters. One of the main uses of the character variety is as a bookkeeping device for hyperbolic Dehn surgeries. The talk's goal is to show how my recent work with Eric Chesebro and Priyadip Mondal uses this property together with some baby complex analysis to give a criterion for answering a certain question about particular hyperbolic knot complements. I promise to define and attempt to motivate all terms specific to the study of 3-manifolds.