Skip to main content
SpringerLink
Log in

Some families of links with divergent Mahler measure

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. Biggs, N.: Equimodluar Cures CDAM Research Report Series. LSE-CDAM-2000-17 (2000)

  3. Bollobas B., Riordan O.: AA tutte polynomial for colored graphs. Comb. Probab. Comput. 8, 45–93 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Champanerkar A., Kofman I.: On the Mahler measure of Jones polynomials under twisting. Algebraic Geom. Topol. 5, 1–22 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  6. Jin X., Zhang F.: The replacements of signed graphs and Kauffman brackets of links. Adv. Appl. Math. 39(2), 155–172 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jin X., Zhang F.: Zeros of the Jones polynomial for families of pretzel links. Phys. A 328, 391–408 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Silver D., Stoimenow A., Williams S.: Euclidean Mahler measure and twisted links. Algebraic Geom. Topol. 6, 581–602 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Silver, D., Williams, S.: Mahler Measure of the Alexander polynomial. J. Lond. Math. Soc., 69(3), 767–782

  10. Sokal A.D.: Chromatic roots are dense in the whole complex plane. Combin. Probab. Comput. 13, 221 (2004) (cond-mat/0012369)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wu, F.Y., Wang, J.: Zeroes of the Jones polynomial. Phys. A 296(3), 483–494(12) (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Nebraska at Omaha, Omaha, NE, 68182, USA

    Robert G. Todd

Authors
  1. Robert G. Todd
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert G. Todd.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Todd, R.G. Some families of links with divergent Mahler measure. Geom Dedicata 159, 337–351 (2012). https://doi.org/10.1007/s10711-011-9663-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-011-9663-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation