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

Fumikazu Nagasato, Yoshikazu Yamaguchi*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


Let K be a knot in an integral homology 3-sphere Σ with exterior EK, 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(EK) 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(EK). 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 languageEnglish
Pages (from-to)967-1002
Number of pages36
JournalMathematische Annalen
Issue number3
Publication statusPublished - 2012 Nov
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)


Dive into the research topics of 'On the geometry of the slice of trace-free SL2(ℂ)-characters of a knot group'. Together they form a unique fingerprint.

Cite this