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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: yfukumot@fc.ritsumei.ac.jp
PAUL KIRK
Affiliation:
Indiana University, Department of Mathematics Bloomington, Indiana 47401, U.S.A. e-mail: pkirk@indiana.edu
JUANITA PINZÓN-CAICEDO
Affiliation:
University of Georgia, Department of Mathematics Athens, GA 30602, U.S.A. e-mail: jpinzon@uga.edu
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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 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.

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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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