Geometry of $\mathrm{SU}(3)$-character varieties of torus knots

We describe the geometry of the character variety of representations of the knot group $\Gamma_{m,n}=\langle x,y| x^n=y^m\rangle$ into the group $\mathrm{SU}(3)$, by stratifying the character variety into strata correspoding to totally reducible representations, representations decomposing into a $2$-dimensional and a $1$-dimensional representation, and irreducible representations, the latter of two types depending on whether the matrices have distinct eigenvalues, or one of the matrices has one eigenvalue of multiplicity $2$. We describe how the closure of each stratum meets lower strata, and use this to compute the compactly supported Euler characteristic, and to prove that the inclusion of the character variety for $\mathrm{SU}(3)$ into the character variety for $\mathrm{SL}(3,\mathbb{C})$ is a homotopy equivalence.