## Character varieties of hyperbolic knot complements

- Date: 03/13/2008
- Time: 16:10

Melissa Macasieb (UBC)

Simon Fraser University

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.

Number Theory Seminar