  We compute both natural and smooth models for the $SL_2(\mathbb C)$ character
varieties of the two component double twist links, an infinite family of
two-bridge links indexed as $J(k,l)$. For each $J(k,l)$, the component(s) of
the character variety containing characters of irreducible representations are
birational to a surface of the form $C\times \mathbb C$ where $C$ is a curve.
The same is true of the canonical component. We compute the genus of this
curve, and the degree of irrationality of the canonical component. We realize
the natural model of the canonical component of the $SL_2(\mathbb C)$ character
variety of the $J(3,2m+1)$ links as the surface obtained from
$\mathbb{P}^1\times \mathbb{P}^1$ as a series of blow-ups.

Geometric Topology