Advertisement

Geometriae Dedicata

, Volume 159, Issue 1, pp 337–351 | Cite as

Some families of links with divergent Mahler measure

  • Robert G. Todd
Original Paper

Abstract

In the following note we develop a method to prove that the Mahler Measure of the Jones polynomial of a family of links diverges. We apply this to several examples from the literature. We then use the W-polynomial to find the Kauffman Bracket of some families of Montesinos links and show that their Jones polynomials too have divergent Mahler measure.

Keywords

Kauffman bracket Jones polynomial Mahler measure 

Mathematics Subject Classification (2000)

57M25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beraha S., Kahane J., Wiess N.J.: Limits of zeros of recursively defined families of polynomials. In: Rota, G.-C. (eds) Studies in Foundations and Combi- natorics (Advances in Mathematics Supplementary Studies, Vol. 1), Academic Press, New York (1978)Google Scholar
  2. 2.
    Biggs, N.: Equimodluar Cures CDAM Research Report Series. LSE-CDAM-2000-17 (2000)Google Scholar
  3. 3.
    Bollobas B., Riordan O.: AA tutte polynomial for colored graphs. Comb. Probab. Comput. 8, 45–93 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Champanerkar A., Kofman I.: On the Mahler measure of Jones polynomials under twisting. Algebraic Geom. Topol. 5, 1–22 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chang S.-C., Shrock R.: Zeros of Jones polynomials for families of Knots and Links. Phys. A Stat. Mech. Appl. 296, 483–494 (2001)CrossRefGoogle Scholar
  6. 6.
    Jin X., Zhang F.: The replacements of signed graphs and Kauffman brackets of links. Adv. Appl. Math. 39(2), 155–172 (2007)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Jin X., Zhang F.: Zeros of the Jones polynomial for families of pretzel links. Phys. A 328, 391–408 (2003)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Silver D., Stoimenow A., Williams S.: Euclidean Mahler measure and twisted links. Algebraic Geom. Topol. 6, 581–602 (2006)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Silver, D., Williams, S.: Mahler Measure of the Alexander polynomial. J. Lond. Math. Soc., 69(3), 767–782Google Scholar
  10. 10.
    Sokal A.D.: Chromatic roots are dense in the whole complex plane. Combin. Probab. Comput. 13, 221 (2004) (cond-mat/0012369)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Wu, F.Y., Wang, J.: Zeroes of the Jones polynomial. Phys. A 296(3), 483–494(12) (2001)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Nebraska at OmahaOmahaUSA

Personalised recommendations