Asymptotics of the colored Jones function of a knot

Garoufalidis, Stavros; Le, Thang T. Q.

doi:10.2140/gt.2011.15.2135

Cited by 54 publications

(42 citation statements)

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“…Using a linear recursion for K ,k (q), it is easy to see that mindeg q ( K ,k (q)) is bounded below by a quadratic function of k; see for example [53,Thm.10.3]. A stronger statement http://www.resmathsci.com/content/2/1/1 is known [4] Lemma 19.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Nahm sums, stability and the colored Jones polynomial

Garoufalidis

2015

Res Math Sci

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114

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Nahm sums are q-series of a special hypergeometric type that appear in character formulas in the conformal field theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in the quantum knot theory -we prove the stability of the coefficients of the colored Jones polynomial of an alternating link and present a Nahm sum formula for the resulting power series, defined in terms of a reduced diagram of the alternating link. The Nahm sum formula comes with a computer implementation, illustrated in numerous examples of proven or conjectural identities among q-series. MSC: Primary 57N10; Secondary 57M25.Keywords: Nahm sums; Colored Jones polynomial; Links; Stability; Modular forms; Mock-modular forms; q-holonomic sequence; q-series; Conformal field theory; Thin-thick decomposition BackgroundThe colored Jones polynomial of a link is a sequence of Laurent polynomials in one variable with integer coefficients. We prove in full a conjecture concerning the stability of the colored Jones polynomial for all alternating links.A weaker form of stability (zero stability, defined below) for the colored Jones polynomial of an alternating knot was conjectured by Dasbach and Lin. The zero stability is also proven independently by Armond for all adequate links [1], which include alternating links and closures of positive braids, see also [2]. The advantage of our approach is that it proves stability to all orders and gives explicit formulas (in the form of generalized Nahm sums) for the limiting series, which in particular implies convergence in the open unit disk in the q-plane and allow for the study of their redial asymptotics.Stability was observed in some examples by Zagier, and conjectured by the first author to hold for all knots, assuming that we restrict the sequence of colored Jones polynomials to suitable arithmetic progressions, dictated by the quasi-polynomial nature of its q-degree [3,4]. Zagier asked about modular and asymptotic properties of the limiting q-series. In a similar direction, Habiro asked about zero stability of the cyclotomic function of alternating links in [5].Our generalized Nahm sum formula comes with a computer implementation (using as input a planar diagram of a link), and allows the study of its asymptotics when q

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Section: Proof Of Theoremmentioning

confidence: 99%

Nahm sums, stability and the colored Jones polynomial

Garoufalidis

2015

Res Math Sci

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114

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“…The above conjecture is a statement about formal power series. A stronger analytic version is known [26,Thm.1.3]; namely, for every knot K there exists an open neighborhood U K of 0 ∈ C such that for all α ∈ U K we have…”

Section: The Colored Jones Polynomial and The Alexander Polynomialmentioning

confidence: 99%

“…More is known about the summation of the series (1) along a fixed diagonal i = j + k for fixed k, both on the level of formal power series and on the analytic counterpart. For further details, the reader may consult [26] and references therein.…”

Section: The Colored Jones Polynomial and The Alexander Polynomialmentioning

confidence: 99%

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Quantum knot invariants

Garoufalidis

2018

Res Math Sci

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This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective computations. The talk was given in the Mathematische Arbeitstagung June 24-July 1, 2011.

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“…Reshetikhin and Turaev [14] defined the quantum SU (2) invariant, and the SO(3) version was defined by Kirby and Melvin [7]. The Ohtsuki invariants was constructed from the quantum SO (3) invariant. Afterwards, in [4,5], Habiro introduced the cyclotomic expansion of the colored Jones polynomial and constructed a unified invariant of integral homology 3-spheres, which is known as the cyclotomic expansion of the Ohtsuki invariant.…”

Section: Introductionmentioning

confidence: 99%

Ohtsuki Invariants for Integral Homology Spheres and Habiro's Cyclotomic Expansion

Takata

2009

J. Knot Theory Ramifications

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We give surgery formulas for the Ohtsuki invariants λ 1 , λ 2 and λ 3 of an integral homology sphere obtained by surgery from S 3 along a knot, related to Habiro's cyclotomic expansion of the colored Jones polynomial of the knot. As an application, we prove that the Ohtsuki invariants λ 1 , λ 2 and λ 3 separate integral homology 3-spheres obtained from S 3 by surgery along the Borromean rings. Theorem 1.1. Let m, s, t be integers and M m,s,t be the integral homology 3sphere obtained from S 3 by surgery along the Borromean rings with framings −1/m, −1/s, −1/t on 3 components, respectively. The Ohtsuki invariants λ 1 , λ 2 and λ 3 separate M m,s,t 's. 21 J. Knot Theory Ramifications 2009.18:21-31. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/04/15. For personal use only. J. Knot Theory Ramifications 2009.18:21-31. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/04/15. For personal use only.

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Asymptotics of the colored Jones function of a knot

Cited by 54 publications

References 32 publications

Nahm sums, stability and the colored Jones polynomial

Nahm sums, stability and the colored Jones polynomial

Quantum knot invariants

Ohtsuki Invariants for Integral Homology Spheres and Habiro's Cyclotomic Expansion

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