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

TL;DR

This paper analyzes the geometric structure of the $ ext{SU}(3)$ character variety for torus knots, classifying representations into strata and computing topological invariants, revealing homotopy equivalences with complex special linear group character varieties.

Contribution

It provides a detailed stratification of the $ ext{SU}(3)$ character variety for torus knots and establishes its topological relationship with $ ext{SL}(3,b{C})$ character varieties.

Findings

Stratification of the character variety into reducible and irreducible components.

Explicit description of how strata closures intersect.

Computation of the compactly supported Euler characteristic.

Abstract

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.