Character varieties of double twist links

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.