## Geometry & Topology

### Asymptotics of the colored Jones function of a knot

#### 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 $eα∕n$, when $α$ is a fixed complex number and $n$ tends to infinity. We analyze this asymptotic behavior to all orders in $1∕n$ 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
Revised: 31 August 2011
Accepted: 4 October 2011
First available in Project Euclid: 20 December 2017

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

#### 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

#### 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