On the geometry of the slice of trace-free SL₂(ℂ)-characters of a knot group

Let K be a knot in an integral homology 3-sphere Σ with exterior E_K, and let B₂ denote the two-fold branched cover of Σ branched along K. We construct a map Φ from the slice of trace-free SL₂(ℂ)-characters of π₁(E_K) to the SL₂(ℂ)-character variety of π₁(B₂). When this map is surjective, it describes the slice as the two-fold branched cover over the SL₂(ℂ)-character variety of B₂ with branched locus given by the abelian characters, whose preimage is precisely the set of metabelian characters. We show that each metabelian character can be represented as the character of a binary dihedral representation of π₁(E_K). The map Φ is shown to be surjective for all 2-bridge knots and all pretzel knots of type (p, q, r). An extension of this framework to n-fold branched covers is also described.