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

Abstract

Let $G$ be the fundamental group of the complement of the torus knot of type $\left(m,n\right)$. It has a presentation $G=\langle x,y\mid x^m=y^n\rangle$. 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 $PGL\left(3,ℂ\right)$.