On the geometry of the slice of trace-free ${{SL_2(\mathbb{C})}}$-characters of a knot group

Abstract

Let K be a knot in an integral homology 3-sphere Σ with exterior E _K , and let B ₂ denote the two-fold branched cover of Σ branched along K. We construct a map Φ from the slice of trace-free ${{{\rm SL}_2(\mathbb{C})}}$ -characters of π ₁(E _K ) to the ${{{\rm SL}_2(\mathbb{C})}}$-character variety of π ₁(B ₂). When this map is surjective, it describes the slice as the two-fold branched cover over the ${{{\rm SL}_2(\mathbb{C})}}$-character variety of B ₂ 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 π ₁(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.