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: 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 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

References:
  1. D. Baraglia and P. Hekmati, Arithmetic of singular character varieties and their E-polynomials, Proc. Lond. Math. Soc. 114 (3), 293–332, 2017.
  2. L. A. Borisov, The class of the affine line is a zero divisor in the Grothendieck ring, J. Algebraic Geom. 27, 203–209, 2018.
  3. M. Culler, P. B. Shalen, Varieties of group representations and splitting of 3-manifolds, Ann. of Math. 117 (1), 109–146, 1983.
  4. P. Deligne, Théorie de Hodge II, Publ. Math. I.H.E.S. 40, 5–58, 1971.
  5. P. Deligne, Théorie de Hodge III, Publ. Math. I.H.E.S. 44, 5–77, 1974.
  6. Á. González-Prieto, Pseudo-quotients of algebraic actions and their application to character varieties, arxiv.org/abs/1807.08540v4
  7. Á. González-Prieto, Motivic theory of representation varieties via Topological Quantum Field Theories, arxiv.org/abs/1810.09714v2
  8. Á. González-Prieto, M. Logares and V. Muñoz, A lax monoidal Topological Quantum Field Theory for representation varieties, Bull. Sci. Math. 161, 102871, 2020.
  9. 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.
  10. 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.
  11. Á. González-Prieto and V. Muñoz, Motive of the SL4-character variety of torus knots, arxiv.org/abs/2006.01810
  12. T. Hausel, E. Letelier and F. R. Villegas, Arithmetic harmonic analysis on character and quiver varieties II, Adv. Math. 234, 85–128, 2013.
  13. 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.
  14. T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153, 197–229, 2003.
  15. N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (3), 59–126, 1987.
  16. S. Lawton and V. Muñoz, E-polynomial of the SL(3, ℂ)-character variety of free groups, Pac. J. Math. 282, 173–202, 2016.
  17. M. Logares, V. Muñoz and P. E. Newstead, Hodge polynomials of SL(2, ℂ)-character varieties for curves of small genus, Rev. Mat. Complut. 26, 635–703, 2013.
  18. A. Lubotzky and A. R. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58, 1985.
  19. 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.
  20. J. Martínez, E-polynomials of PGL(2, ℂ)-character varieties of surface groups, arxiv.org/abs/1705.04649.
  21. J. Martínez and V. Muñoz, E-polynomials of the SL(2, ℂ)-character varieties of surface groups, Int. Math. Res. Not. IMRN 2016, 926–961, 2016.
  22. J. Martínez and V. Muñoz, E-polynomial of the SL(2, ℂ)-character variety of a complex curve of genus 3, Osaka J. Math. 53, 645–681, 2016.
  23. M. Mereb, On the E-polynomials of a family of SLn-character varieties, Math. Ann. 363, 857–892, 2015.
  24. V. Muñoz, The SL(2, ℂ)-character varieties of torus knots, Rev. Mat. Complut. 22, 489–497, 2009.
  25. V. Muñoz and J. Porti, Geometry of the SL(3, ℂ)-character variety of torus knots, Algebr. Geom. Topol. 16, 397–426, 2016.
  26. M. Nagata, Invariants of a group in an affine ring, J. Math. Kyoto Univ. 3, 369–377, 1963/1964.
  27. P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 51, TIFR, 1978.
  28. D. Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, 1990.