  Let M be a two cusped hyperbolic 3-manifold and let M(r) be the result of r
Dehn filling of a fixed cusp of M. We study canonical components of the SL(2,C)
character varieties of M(r). We show that the gonality of these sets is
bounded, independent of the filling parameter. We also obtain bounds, depending
on r, for the genus of these sets. We compute the gonality for the double twist
knots, demonstrating canonical components with arbitrarily large gonality.
