Abstract
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.
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References
Adams, W.W., Loustaunau, P.: An introduction to Gröbner bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994)
Burde, G., Zieschang, H.: Knots. de Gruyter Studies in Mathematics, vol. 5. Walter de Gruyter, Berlin, New York (1985)
Crowell, R.H., Fox, R.H.: Introduction to Knot Theory. Graduate Texts in Mathematics, vol. 57. Springer, New York (1977)
Fox, R.H., Smythe, N.: An ideal class invariant of knots. Proc. Amer. Math. Soc. 15, 707–709 (1964)
Kanenobu, T.: Infinitely many knots with the same polynomial invariant. Trans. Amer. Math. Soc. 97, 158–162 (1986)
Kawauchi, A.: A survey of knot theory. Birkhäuser Verlag, Basel (1996)
Kearton, C., Wilson, S.M.J.: Knot modules and the Nakanishi index. Proc. Amer. Math. Soc. 131, 655–663 (2003)
Lickorish, W.B.R.: An introduction to knot theory. Graduate Texts in Mathematics, vol. 175. Springer, New York (1998)
Livingston, C.: Table of knot invariants, At: http://www.indiana.edu/~knotinfo/
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)
Sims, C.C.: Computation with finitely presented groups. Encyclopedia of Mathematics and its Applications, vol. 48. Cambridge University Press, Cambridge (1994)
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© 2006 Springer-Verlag Berlin Heidelberg
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Gago-Vargas, J., Hartillo-Hermoso, I., Ucha-Enríquez, J.M. (2006). Algorithmic Invariants for Alexander Modules. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2006. Lecture Notes in Computer Science, vol 4194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11870814_12
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DOI: https://doi.org/10.1007/11870814_12
Publisher Name: Springer, Berlin, Heidelberg
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