Algebraic & Geometric Topology

On the Mahler measure of Jones polynomials under twisting

Abhijit Champanerkar and Ilya Kofman

Full-text: Open access

Abstract

We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume under Dehn surgery. For pretzel links P(a1,,an), we show that the Mahler measure of the Jones polynomial converges if all ai, and approaches infinity for ai= constant if n, just as hyperbolic volume. We also show that after sufficiently many twists, the coefficient vector of the Jones polynomial and of any colored Jones polynomial decomposes into fixed blocks according to the number of strands twisted.

Article information

Source
Algebr. Geom. Topol., Volume 5, Number 1 (2005), 1-22.

Dates
Received: 13 October 2004
Revised: 6 November 2004
Accepted: 7 December 2004
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796390

Digital Object Identifier
doi:10.2140/agt.2005.5.1

Mathematical Reviews number (MathSciNet)
MR2135542

Zentralblatt MATH identifier
1061.57007

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05]

Keywords
Jones polynomial Mahler measure Temperley–Lieb algebra hyperbolic volume

Citation

Champanerkar, Abhijit; Kofman, Ilya. On the Mahler measure of Jones polynomials under twisting. Algebr. Geom. Topol. 5 (2005), no. 1, 1--22. doi:10.2140/agt.2005.5.1. https://projecteuclid.org/euclid.agt/1513796390


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References

  • D Boyd, F Rodriguez-Villegas, Mahler's measure and the dilogarithm. I, Canad. J. Math. 54 (2002) 468–492
  • D Boyd, F Rodriguez-Villegas, N Dunfield, Mahler's Measure and the Dilogarithm (II).
  • A Champanerkar, I Kofman, E Patterson, The next simplest hyperbolic knots, J. Knot Theory Ramifications 13 (2004) 965–987
  • S Chang, R Shrock, Zeros of Jones polynomials for families of knots and links, Phys. A 301 (2001) 196–218
  • O Dasbach, X-S Lin, A volume-ish theorem for the Jones polynomial of alternating knots.
  • F Goodman, P de la Harpe, V Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications 14, Springer-Verlag, New York (1989)
  • S Gukov, Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial \arxivhep-th/0306165
  • X Jin, F Zhang, Zeros of the Jones polynomials for families of pretzel links, Phys. A 328 (2003) 391–408
  • V Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987) 335–388
  • L Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990) 417–471
  • A Kawauchi, A survey of knot theory, Birkhäuser Verlag, Basel (1996)
  • R Kirby, P Melvin, The $3$-manifold invariants of Witten and Reshetikhin-Turaev for ${\rm sl}(2,{\bf C})$, Invent. Math. 105 (1991) 473–545
  • M Lackenby, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. 88 (2004) 204–224, with an appendix by I Agol and D Thurston
  • R Landvoy, The Jones polynomial of pretzel knots and links, Topology Appl. 83 (1998) 135–147
  • W Lawton, A problem of Boyd concerning geometric means of polynomials, J. Number Theory 16 (1983) 356–362
  • X-S Lin, Zeros of the Jones polynomial,\nl http://math.ucr.edu/~xl/abs-jk.pdf
  • H Murakami, J Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85–104
  • V Prasolov, A Sossinsky, Knots, links, braids and 3-manifolds, Translations of Mathematical Monographs 154, American Mathematical Society, Providence, RI (1997)
  • A Schinzel, The Mahler measure of polynomials, from: “Number theory and its applications (Ankara, 1996)”, Lecture Notes in Pure and Appl. Math. 204, Dekker, New York (1999) 171–183
  • A Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications 77, Cambridge University Press (2000)
  • D Silver, S Williams, Mahler measure of Alexander polynomials, J. London Math. Soc. 69 (2004) 767–782
  • N-P Skoruppa, Heights, Graduate course, Bordeaux (1999)\nl http://wotan.algebra.math.uni-siegen.de/~countnumber/D/
  • C Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971) 169–175
  • H Wenzl, Hecke algebras of type $A\sb n$ and subfactors, Invent. Math. 92 (1988) 349–383
  • F Wu, J Wang, Zeroes of the Jones polynomial, Phys. A 296 (2001) 483–494
  • Y Yokota, Twisting formulas of the Jones polynomial, Math. Proc. Cambridge Philos. Soc. 110 (1991) 473–482