Motive of the $SL_4$-character variety of torus knots

TL;DR

This paper computes the motive of the $SL_4$-character variety for torus knots by stratifying the variety based on a canonical filtration, reducing the problem to combinatorics.

Contribution

It introduces a new stratification method for character varieties, enabling motive computation for $SL_4$ representations of torus knot groups.

Findings

Motive of the $SL_4$-character variety explicitly computed.

Stratification reduces complex geometric problems to combinatorial ones.

Applicable over any algebraically closed field of zero characteristic.

Abstract

In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into $SL_4(k)$, for any algebraically closed field $k$ of zero characteristic. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This allows us to reduce the computation of the motive to a combinatorial problem.