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1 results

  • Traceless SU(2) representations of 2-stranded tangles

  • YOSHIHIRO FUKUMOTO, PAUL KIRK, JUANITA PINZÓN-CAICEDO
  • Journal:
    Mathematical Proceedings of the Cambridge Philosophical Society / Volume 162 / Issue 1 / January 2017
    Published online by Cambridge University Press:
    03 June 2016, pp. 101-129
    Print publication:
    January 2017

  • Given a 2-stranded tangle T contained in a ℤ-homology ball Y, we investigate the character variety R(Y, T) of conjugacy classes of traceless SU(2) representations of π1(Y \ T). In particular we completely determine the subspace of binary dihedral representations, and identify all of R(Y, T) for many tangles naturally associated to knots in S 3. Moreover, we determine the image of the restriction map from R(T, Y) to the traceless SU(2) character variety of the 4-punctured 2-sphere (the pillowcase). We give examples to show this image can be non-linear in general, and show it is linear for tangles associated to pretzel knots.