#### Volume 16, issue 1 (2016)

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Geometry of the ${\rm SL}(3,\mathbb C)$–character variety of torus knots

### Vicente Muñoz and Joan Porti

Algebraic & Geometric Topology 16 (2016) 397–426
##### Abstract

Let $G$ be the fundamental group of the complement of the torus knot of type $\left(m,n\right)$. It has a presentation $G=〈x,y\mid {x}^{m}={y}^{n}〉$. We find a geometric description of the character variety $X\left(G\right)$ of characters of representations of $G$ into $SL\left(3,ℂ\right)$, $GL\left(3,ℂ\right)$ and $PGL\left(3,ℂ\right)$.

##### Keywords
torus knot, characters, representations
##### Mathematical Subject Classification 2010
Primary: 14D20
Secondary: 57M25, 57M27
##### Publication
Received: 29 September 2014
Revised: 7 June 2015
Accepted: 12 June 2015
Published: 23 February 2016
##### Authors
 Vicente Muñoz Facultad de Matemáticas Universidad Complutense de Madrid Plaza de Ciencias 3 28040 Madrid Spain Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) C/ Nicolás Cabrera 15 28049 Madrid Spain http://www.mat.ucm.es/~vmunozve/ Joan Porti Departament de Matemàtiques Universitat Autònoma de Barcelona Cerdanyola del Valles 08193 Barcelona Spain http://mat.uab.es/~porti/