Character varieties of hyperbolic knot complements

To every hyperbolic 3-manifold M with nonempty boundary, one can associate a pair of related algebraic varieties X(M) and Y(M) called the character varieties of M. These varieties carry much topological information about M, but are in general difficult to compute. In the case that M is a knot complement, X(M) and Y(M) are defined over Q. In this talk, I will discuss how properties of these varieties reflect the topology of M in the case M is a hyperbolic knot complement. I will also show how to obtain explicit equations for the the character varieties associated to a bi-infinite family of hyperbolic knots K(m,n) and discuss consequences of these results related to the existence of integral points on these curves. This is joint work with Kate Petersen and Ronald van Luijk.