Algorithmic Invariants for Alexander Modules

  • Jesús Gago-Vargas
  • Isabel Hartillo-Hermoso
  • José María Ucha-Enríquez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4194)


Let G be a group given by generators and relations. It is possible to compute a presentation matrix of a module over a ring through Fox’s differential calculus. We show how to use Gröbner bases as an algorithmic tool to compare the chains of elementary ideals defined by the matrix. We apply this technique to classical examples of groups and to compute the elementary ideals of Alexander matrix of knots up to 11 crossings with the same Alexander polynomial.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Adams et al.(1994)]
    Adams, W.W., Loustaunau, P.: An introduction to Gröbner bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994)zbMATHGoogle Scholar
  2. [Burde et al.(1985)]
    Burde, G., Zieschang, H.: Knots. de Gruyter Studies in Mathematics, vol. 5. Walter de Gruyter, Berlin, New York (1985)zbMATHGoogle Scholar
  3. [Crowell et al.(1977)]
    Crowell, R.H., Fox, R.H.: Introduction to Knot Theory. Graduate Texts in Mathematics, vol. 57. Springer, New York (1977)zbMATHGoogle Scholar
  4. [Fox et al.(1964)]
    Fox, R.H., Smythe, N.: An ideal class invariant of knots. Proc. Amer. Math. Soc. 15, 707–709 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Kanenobu(1986)]
    Kanenobu, T.: Infinitely many knots with the same polynomial invariant. Trans. Amer. Math. Soc. 97, 158–162 (1986)zbMATHMathSciNetGoogle Scholar
  6. [Kawauchi(1996)]
    Kawauchi, A.: A survey of knot theory. Birkhäuser Verlag, Basel (1996)zbMATHGoogle Scholar
  7. [Kearton et al.(2003)]
    Kearton, C., Wilson, S.M.J.: Knot modules and the Nakanishi index. Proc. Amer. Math. Soc. 131, 655–663 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Lickorish(1998)]
    Lickorish, W.B.R.: An introduction to knot theory. Graduate Texts in Mathematics, vol. 175. Springer, New York (1998)Google Scholar
  9. [Livingston(2004)]
    Livingston, C.: Table of knot invariants, At:
  10. [Pauer et al.(1999)]
    Pauer, F., Unterkircher, A.: Gröbner Bases for Ideals in Laurent Polynomials Rings and their Application to Systems of Difference Equations. Appl. Algebra Engrg. Comm. Comput. 9, 271–291 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Sims(1994)]
    Sims, C.C.: Computation with finitely presented groups. Encyclopedia of Mathematics and its Applications, vol. 48. Cambridge University Press, Cambridge (1994)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jesús Gago-Vargas
    • 1
  • Isabel Hartillo-Hermoso
    • 2
  • José María Ucha-Enríquez
    • 1
  1. 1.Depto. de ÁlgebraUniv. de SevillaSevillaSpain
  2. 2.Dpto. de MatemáticasUniv. de CádizPuerto Real, CádizSpain

Personalised recommendations