The Cusp Parameter Function on a Hyperbolic 3-Manifold Deformation Variety: An Example

The deformation variety of a hyperbolic 3-manifold M is an affine algebraic set that parametrizes the complete and incomplete hyperbolic structures on M. When M is non-compact but of finite volume the non-compactness occurs in regions called cusps, and for each cusp c the deformation variety has a codimension-one algebraic subset consisting of (possibly incomplete) hyperbolic structures where c remains complete.

In this talk I’ll ground those definitions with a particular example, the “6_2^2 link complement”, and further explain how answering a geometric question about this manifold comes down to understanding certain imaginary quadratic values of a rational function that records geometric data associated to a particular cusp.