On the twisted Alexander polynomial for metabelian represetations into $SL_{2}(\mathbb{C

We devote this note to expose an explicit form of the twisted Alexander invariant for irreducible metabelian $SL_{2}(\mathbb{C})$-representations of knot groups. This work was mo-tivated by the characterization of irreducible metabelian $SL_{2}(\mathbb{C}) $-representations in [9], concerning the conjugacy classes of $SL_{2}(\mathbb{C})$-representations. We can correspond the se