## Illinois Journal of Mathematics

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

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

Michael Heusener, Vicente Muñoz, and Joan Porti

#### 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**

Received: 11 August 2015

Revised: 25 May 2016

First available in Project Euclid: 21 June 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/ijm/1498032024

**Mathematical Reviews number (MathSciNet)**

MR3665172

**Zentralblatt MATH identifier**

1373.57014

**Subjects**

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}

Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

#### 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