Stratification of $\mathrm{AGL}_r(\mathbb{C})$-representation varieties of twisted Hopf links

Abstract

We provide a stratification of the $\mathrm{AGL}_r(\mathbb{C})$-representation variety of the fundamental group of the complement of a twisted Hopf link in terms of a stratification of the corresponding $\mathrm{GL}_r(\mathbb{C})$-representation variety. For ranks $1$ and $2$, we explicitly describe this stratification and compute the motives of these varieties in terms of the Lefschetz motive $q=[\mathbb{C}]$ in the Grothendieck ring of complex algebraic varieties $K_0(\mathbf{Var}_{\mathbb{C}})$.