Searching for the cusp
We will review the details of algebraically finding cusps on the boundaries of deformation spaces of kleinian groups. The geometric starting point is an arrangement of circles with prescribed tangencies and relationships under a set of M\"obius transformations. These lead to polynomial equations in several complex variables, which may then be numerically solved for the values which describe the cusp. We will go through this process for several deformation spaces corresponding to "plumbing" constructions of Maskit and Kra, and we will present some of the numerical output. The same techniques can also be used to calculate the coherent spiral hexagonal circle packings discovered by Peter Doyle, and we will compare the similarity factors of those packings to cusps on boundaries of deformation spaces.