  Let K be a knot in an integral homology 3-sphere and let B denote the 2-fold
branched cover of the integral homology sphere branched along K. We construct a
map from the slice of characters with trace free along meridians in the SL(2,
C)-character variety of the knot exterior to the SL(2, C)-character variety of
2-fold branched cover B. When this map is surjective, it describes the slice as
the 2-fold branched cover over the SL(2, C)-character variety of B with
branched locus given by the abelian characters, whose preimage is precisely the
set of metabelian characters. We show that each of metabelian character can be
represented as the character of a binary dihedral representation of the knot
group. This map 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.
