Article ID: MTJPAM-D-21-00033

Title: The Point Counting Problem in Representation Varieties of Torus Knots

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00033; Volume 4 / Issue 3 / Year 2022 (Special Issue), Pages 114-130

Document Type: Research Paper

Author(s): Ángel González-Prieto a , Vicente Muñoz b

aDepartamento de Álgebra, Geometría y Topología, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain

bDepartamento de Álgebra, Geometría y Topología, Universidad de Málaga, Campus de Teatinos s/n, 29071 Málaga, Spain

Received: 19 May 2021, Accepted: 4 February 2022, Published: 4 April 2022.

Corresponding Author: Vicente Muñoz (Email address:

Full Text: PDF


We compute the motive of the variety of representations of the torus knot of type (m, n) into the affine groups AGL1(k) and AGL2(k) for an arbitrary field k. In the case that k = 𝔽q 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.

Keywords: Torus knots, Representation varieties, Affine group, Finite fields

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