On the geometry of the slice of trace--free SL(2,C)-characters of a knot group

TL;DR

This paper explores the structure of trace-free SL(2,C)-characters of knot groups, establishing a map to branched covers that is surjective for certain knots and linking metabelian characters to binary dihedral representations.

Contribution

It constructs a new map connecting trace-free character slices of knot groups to branched covers, revealing surjectivity for specific classes and characterizing metabelian characters.

Findings

The map is surjective for 2-bridge and certain pretzel knots.

Metabelian characters correspond to binary dihedral representations.

Framework extends to n-fold branched covers.

Abstract

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.