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

TL;DR

This paper stratifies the representation varieties of twisted Hopf links for the group $ ext{AGL}_r( ext{C})$, explicitly describes these stratifications for ranks 1 and 2, and computes their motives.

Contribution

It introduces a stratification approach for these varieties and explicitly computes motives for low ranks, advancing understanding of their algebraic structure.

Findings

Explicit stratification for ranks 1 and 2

Motivic computations in the Grothendieck ring

Connection between $ ext{AGL}_r( ext{C})$ and $ ext{GL}_r( ext{C})$ representations

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 $K_0(\mathbf{Var}_{\mathbb{C}})$.