Illinois Journal of Mathematics

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

Michael Heusener, Vicente Muñoz, and Joan Porti

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

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


Export citation