The $\mathrm{S}\mathrm{L}(2,\mathbb{C})$-character variety of the magic $3$-manifold

We determine the irreducible $\mathrm{S}\mathrm{L}(2,\mathbb{C})$-character variety of the 3-chain link exterior which is called the `magic $3$-manifold’, and deduce a formula for the twisted Alexander polynomial associated to each $\mathrm{S}\mathrm{L}(2,\mathbb{C})$-representation.