Open Access
Spring 2016 The $SL(3,\mathbb{C})$-character variety of the figure eight knot
Michael Heusener, Vicente Muñoz, Joan Porti
Illinois J. Math. 60(1): 55-98 (Spring 2016). DOI: 10.1215/ijm/1498032024


We give explicit equations that describe the character variety of the figure eight knot for the groups $\mathrm{SL}(3,\mathbb{C})$, $\mathrm{GL}(3,\mathbb{C})$ and $\mathrm{PGL}(3,\mathbb{C})$. For any of these $G$, it has five components of dimension $2$, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from $\mathrm{SL}(2,\mathbb{C})$. The other two components are induced by exceptional Dehn fillings of the figure eight knot. We also describe the action of the symmetry group of the figure eight knot on the character varieties.


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Michael Heusener. Vicente Muñoz. Joan Porti. "The $SL(3,\mathbb{C})$-character variety of the figure eight knot." Illinois J. Math. 60 (1) 55 - 98, Spring 2016.


Received: 11 August 2015; Revised: 25 May 2016; Published: Spring 2016
First available in Project Euclid: 21 June 2017

zbMATH: 1373.57014
MathSciNet: MR3665172
Digital Object Identifier: 10.1215/ijm/1498032024

Primary: 14D20
Secondary: 57M25 , 57M27

Rights: Copyright © 2016 University of Illinois at Urbana-Champaign

Vol.60 • No. 1 • Spring 2016
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