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We compute the motive of the variety of representations of the torus knot oftype (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 countof the number of points of the representation variety, while for $k = C$ thiscalculation returns the E-polynomial of the representation variety. We discussthe interplay between these two results in sight of Katz theorem that relatesthe point count polynomial with the E-polynomial. In particular, we shall showthat 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.

The point counting problem in representation varieties of torus knots