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Traceless SU(2) representations of 2-stranded tangles

Published online by Cambridge University Press:  03 June 2016

YOSHIHIRO FUKUMOTO ,
PAUL KIRK  and
JUANITA PINZÓN-CAICEDO
YOSHIHIRO FUKUMOTO
Affiliation:
Ritsumeikan University, Department of Mathematical Sciences 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan. e-mail: [email protected]
PAUL KIRK
Affiliation:
Indiana University, Department of Mathematics Bloomington, Indiana 47401, U.S.A. e-mail: [email protected]
JUANITA PINZÓN-CAICEDO
Affiliation:
University of Georgia, Department of Mathematics Athens, GA 30602, U.S.A. e-mail: [email protected]
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Abstract

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 S3. 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 1 , January 2017 , pp. 101 - 129
Copyright
Copyright © Cambridge Philosophical Society 2016 

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