**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}

^{a}Departamento de Álgebra, Geometría y Topología, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain

^{b}Departamento 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: vicente.munoz@uma.es)

**Full Text:** PDF

**Abstract**

We compute the motive of the variety of representations of the torus knot of type (*m*, *n*) into the affine groups *A**G**L*_{1}(**k**) and *A**G**L*_{2}(**k**) 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.

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

**References:**

- D. Baraglia and P. Hekmati,
*Arithmetic of singular character varieties and their*, Proc. Lond. Math. Soc.*E*-polynomials**114 (3)**, 293–332, 2017. - L. A. Borisov,
*The class of the affine line is a zero divisor in the Grothendieck ring*, J. Algebraic Geom.**27**, 203–209, 2018. - M. Culler, P. B. Shalen,
*Varieties of group representations and splitting of 3-manifolds*, Ann. of Math.**117 (1)**, 109–146, 1983. - P. Deligne,
*Théorie de Hodge II*, Publ. Math. I.H.E.S.**40**, 5–58, 1971. - P. Deligne,
*Théorie de Hodge III*, Publ. Math. I.H.E.S.**44**, 5–77, 1974. - Á. González-Prieto,
*Pseudo-quotients of algebraic actions and their application to character varieties*, arxiv.org/abs/1807.08540v4 - Á. González-Prieto,
*Motivic theory of representation varieties via Topological Quantum Field Theories*, arxiv.org/abs/1810.09714v2 - Á. González-Prieto, M. Logares and V. Muñoz,
*A lax monoidal Topological Quantum Field Theory for representation varieties*, Bull. Sci. Math.**161**, 102871, 2020. - Á. González-Prieto, M. Logares and V. Muñoz,
*Representation variety for the rank one affine group*, In eds. I. N. Parasidis, E. Providas and Th. M. Rassias,*Mathematical Analysis in Interdisciplinary Research*, Springer, to appear. - Á. González-Prieto, M. Logares and V. Muñoz,
*Motive of the representation varieties of torus knots for low rank affine groups*, In eds. P. Pardalos and Th. M. Rassias,*Analysis, Geometry, Nonlinear Optimization and Applications*, World Scientific Publ. Co., to appear. - Á. González-Prieto and V. Muñoz,
*Motive of the*, arxiv.org/abs/2006.01810*S**L*_{4}-character variety of torus knots - T. Hausel, E. Letelier and F. R. Villegas,
*Arithmetic harmonic analysis on character and quiver varieties II*, Adv. Math.**234**, 85–128, 2013. - T. Hausel and F. Rodríguez-Villegas,
*Mixed Hodge polynomials of character varieties,*With an appendix by Nicholas M. Katz, Invent. Math.**174**, 555–624, 2008. - T. Hausel and M. Thaddeus,
*Mirror symmetry, Langlands duality, and the Hitchin system*, Invent. Math.**153**, 197–229, 2003. - N.J. Hitchin,
*The self-duality equations on a Riemann surface*, Proc. London Math. Soc.**55 (3)**, 59–126, 1987. - S. Lawton and V. Muñoz,
*E-polynomial of the*, Pac. J. Math.*S**L*(3, ℂ)-character variety of free groups**282**, 173–202, 2016. - M. Logares, V. Muñoz and P. E. Newstead,
*Hodge polynomials of*, Rev. Mat. Complut.*S**L*(2, ℂ)-character varieties for curves of small genus**26**, 635–703, 2013. - A. Lubotzky and A. R. Magid,
*Varieties of representations of finitely generated groups*, Mem. Amer. Math. Soc.**58**, 1985. - J. Martín-Morales and A. M. Oller-Marcén,
*Combinatorial aspects of the character variety of a family of one-relator groups*, Topology Appl.**156**, 2376–2389, 2009. - J. Martínez,
*E-polynomials of*, arxiv.org/abs/1705.04649.*P**G**L*(2, ℂ)-character varieties of surface groups - J. Martínez and V. Muñoz,
*E-polynomials of the*, Int. Math. Res. Not. IMRN*S**L*(2, ℂ)-character varieties of surface groups**2016**, 926–961, 2016. - J. Martínez and V. Muñoz,
*E-polynomial of the*, Osaka J. Math.*S**L*(2, ℂ)-character variety of a complex curve of genus 3**53**, 645–681, 2016. - M. Mereb,
*On the*, Math. Ann.*E*-polynomials of a family of*S**L*_{n}-character varieties**363**, 857–892, 2015. - V. Muñoz,
*The*, Rev. Mat. Complut.*S**L*(2, ℂ)-character varieties of torus knots**22**, 489–497, 2009. - V. Muñoz and J. Porti,
*Geometry of the*, Algebr. Geom. Topol.*S**L*(3, ℂ)-character variety of torus knots**16**, 397–426, 2016. - M. Nagata,
*Invariants of a group in an affine ring*, J. Math. Kyoto Univ.**3**, 369–377, 1963/1964. - P. E. Newstead,
*Introduction to moduli problems and orbit spaces*, Tata Institute of Fundamental Research Lectures on Mathematics and Physics**51**, TIFR, 1978. - D. Rolfsen,
*Knots and links*, Mathematics Lecture Series**7**, Publish or Perish, 1990.