The point counting problem in representation varieties of torus knots

TL;DR

This paper computes the motives and point counts of representation varieties of torus knots into affine groups, revealing multiple point count polynomials and their relation to the E-polynomial, depending on field characteristics.

Contribution

It introduces explicit motive calculations for torus knot representations into affine groups and explores their connection to point counts and E-polynomials across different fields.

Findings

Multiple point count polynomials depend on m, n, and field characteristic.

Only one point count polynomial matches the E-polynomial.

The work links arithmetic properties with geometric invariants.

Abstract

We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1$ and $AGL_2$ for an arbitrary field $k$. In the case that $k = F_q$ is a finite field this gives rise to the count of the number of points of the representation variety, while for $k = C$ this calculation returns the E-polynomial of the representation variety. We discuss the interplay between these two results in sight of Katz theorem that relates the point count polynomial with the E-polynomial. In particular, we shall show that several point count polynomials exist for these representation varieties, depending on the arithmetic between m,n and the characteristic of the field, whereas only one of them agrees with the actual E-polynomial.