Motive of the representation varieties of torus knots for low rank affine groups

TL;DR

This paper calculates the motives of representation varieties of torus knots into low-rank affine groups, revealing their structure within a specific algebraic subring, thus advancing understanding of knot group representations.

Contribution

It provides explicit motive computations for torus knot representations into AGL_1(C) and AGL_2(C), introducing stratification techniques and algebraic insights.

Findings

Motives of representation varieties are computed explicitly.

The motives are contained within a subring generated by the Lefschetz motive.

Stratification methods are used to analyze the varieties.

Abstract

We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1(C)$ and $AGL_2(C)$. For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q=[C].