Geometry & Topology

Asymptotics of the colored Jones function of a knot

Stavros Garoufalidis and Thang T Q Lê

Full-text: Open access

Abstract

To a knot in 3–space, one can associate a sequence of Laurent polynomials, whose n–th term is the n–th colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the n–th colored Jones polynomial at eαn, when α is a fixed complex number and n tends to infinity. We analyze this asymptotic behavior to all orders in 1n when α is a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of the n–th colored Jones polynomial, with applications to upper bounds in the Generalized Volume Conjecture. Work of Agol, Dunfield, Storm and W Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when α is near 2πi. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.

Article information

Source
Geom. Topol., Volume 15, Number 4 (2011), 2135-2180.

Dates
Received: 27 September 2007
Revised: 31 August 2011
Accepted: 4 October 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732364

Digital Object Identifier
doi:10.2140/gt.2011.15.2135

Mathematical Reviews number (MathSciNet)
MR2860990

Zentralblatt MATH identifier
1239.57029

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
hyperbolic volume conjecture colored Jones function Jones polynomial $R$–matrices regular ideal octahedron weave hyperbolic geometry Catalan's constant Borromean rings cyclotomic expansion loop expansion asymptotic expansion WKB $q$–difference equations perturbation theory Kontsevich integral

Citation

Garoufalidis, Stavros; Lê, Thang T Q. Asymptotics of the colored Jones function of a knot. Geom. Topol. 15 (2011), no. 4, 2135--2180. doi:10.2140/gt.2011.15.2135. https://projecteuclid.org/euclid.gt/1513732364


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References

  • I Agol, P A Storm, W P Thurston, Lower bounds on volumes of hyperbolic Haken $3$-manifolds, J. Amer. Math. Soc. 20 (2007) 1053–1077 With an appendix by N Dunfield
  • G E Andrews, The theory of partitions, Cambridge Math. Library, Cambridge Univ. Press (1998) Reprint of the 1976 original
  • D Bar-Natan, S Garoufalidis, On the Melvin–Morton–Rozansky conjecture, Invent. Math. 125 (1996) 103–133
  • D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of $3$–manifolds, Invent. Math. 118 (1994) 47–84
  • N Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999) 623–657
  • J L Dupont, C K Zickert, A dilogarithmic formula for the Cheeger–Chern–Simons class, Geom. Topol. 10 (2006) 1347–1372
  • S Garoufalidis, Beads: from Lie algebras to Lie groups
  • S Garoufalidis, J S Geronimo, Asymptotics of $q$–difference equations, from: “Primes and knots”, (T Kohno, M Morishita, editors), Contemp. Math. 416, Amer. Math. Soc. (2006) 83–114
  • S Garoufalidis, A Kricker, A rational noncommutative invariant of boundary links, Geom. Topol. 8 (2004) 115–204
  • S Garoufalidis, T T Q Lê, The colored Jones function is $q$–holonomic, Geom. Topol. 9 (2005) 1253–1293
  • S Garoufalidis, L Rozansky, The loop expansion of the Kontsevich integral, the null-move and $S$–equivalence, Topology 43 (2004) 1183–1210
  • S Garoufalidis, X Sun, The $C$–polynomial of a knot, Algebr. Geom. Topol. 6 (2006) 1623–1653
  • S Gukov, Three-dimensional quantum gravity, Chern–Simons theory, and the A-polynomial, Comm. Math. Phys. 255 (2005) 577–627
  • K Habiro, On the colored Jones polynomials of some simple links, from: “Recent progress towards the volume conjecture (Japanese) (Kyoto, 2000)”, Sūrikaisekikenkyūsho Kōkyūroku 1172 (2000) 34–43
  • K Habiro, A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres, Invent. Math. 171 (2008) 1–81
  • E Hille, Analytic function theory, Vol. II, Intro. to Higher Math., Ginn and Co., Boston-New York-Toronto (1962)
  • V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. $(2)$ 126 (1987) 335–388
  • R M Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269–275
  • 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. $(3)$ 88 (2004) 204–224 With an appendix by I Agol and D Thurston
  • T T Q Lê, Integrality and symmetry of quantum link invariants, Duke Math. J. 102 (2000) 273–306
  • T T Q Lê, The colored Jones polynomial and the $A$–polynomial of knots, Adv. Math. 207 (2006) 782–804
  • K Mahler, An application of Jensen's formula to polynomials, Mathematika 7 (1960) 98–100
  • H Murakami, Some limits of the colored Jones polynomials of the figure-eight knot, Kyungpook Math. J. 44 (2004) 369–383
  • H Murakami, J Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85–104
  • F W J Olver, Asymptotics and special functions, AKP Classics, A K Peters Ltd., Wellesley, MA (1997) Reprint of the 1974 original
  • J G Ratcliffe, Foundations of hyperbolic manifolds, second edition, Graduate Texts in Math. 149, Springer, New York (2006)
  • N Y Reshetikhin, V G Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1–26
  • L Rozansky, A universal $U(1)$–rcc invariant of links and rationality conjecture
  • L Rozansky, The universal $R$–matrix, Burau representation, and the Melvin–Morton expansion of the colored Jones polynomial, Adv. Math. 134 (1998) 1–31
  • J L Schiff, Normal families, Universitext, Springer, New York (1993)
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m/ {\unhbox0
  • V G Turaev, Quantum invariants of knots and $3$–manifolds, de Gruyter Studies in Math. 18, de Gruyter, Berlin (1994)
  • R van der Veen, Proof of the volume conjecture for Whitehead chains, Acta Math. Vietnam. 33 (2008) 421–431
  • E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351–399