## Illinois Journal of Mathematics

### The $SL(3,\mathbb{C})$-character variety of the figure eight knot

#### Abstract

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.

#### Article information

Source
Illinois J. Math., Volume 60, Number 1 (2016), 55-98.

Dates
Revised: 25 May 2016
First available in Project Euclid: 21 June 2017

https://projecteuclid.org/euclid.ijm/1498032024

Digital Object Identifier
doi:10.1215/ijm/1498032024

Mathematical Reviews number (MathSciNet)
MR3665172

Zentralblatt MATH identifier
1373.57014

#### Citation

Heusener, Michael; Muñoz, Vicente; Porti, Joan. The $SL(3,\mathbb{C})$-character variety of the figure eight knot. Illinois J. Math. 60 (2016), no. 1, 55--98. doi:10.1215/ijm/1498032024. https://projecteuclid.org/euclid.ijm/1498032024