## 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 SL_{2}(ℂ)-characters of π_{1}(E_{K}) to the SL_{2}(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 SL_{2}(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.

Original language | English |
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Pages (from-to) | 967-1002 |

Number of pages | 36 |

Journal | Mathematische Annalen |

Volume | 354 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 Nov |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)

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