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 $e^{\a/n}$, when $\a$ is a fixed complex number and $n$ tends to infinity. We analyze this asymptotic behavior to all orders in $1/n$ when $\a$ 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-W.Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when $\a$ is near $2\pi i$. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.

31 pages, 13 figures

References in corpus (8)

• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• The colored Jones function is q-holonomic
• A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
• Chern-Simons theory, analytic continuation and arithmetic
• Proof of the volume conjecture for Whitehead chains
• A dilogarithmic formula for the Cheeger-Chern-Simons class
• The C-polynomial of a knot
• Some limits of the colored Jones polynomials of the figure-eight knot

Cited by in corpus (31)

• Analytic Continuation Of Chern-Simons Theory
• SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
• Geometrically and diagrammatically maximal knots
• Analyticity of the Free Energy of a Closed 3-Manifold
• Behavior of knot invariants under genus 2 mutation
• Double affine Hecke algebras and generalized Jones polynomials
• Universal Knot Diagrams
• Planar decomposition of the HOMFLY polynomial for bipartite knots and links
• Quantum Knot Invariants
• On the Volume Conjecture for hyperbolic Dehn-filled $3$-manifolds along the figure-eight knot
• On the volume conjecture for classical spin networks
• Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms II
• Nahm sums, stability and the colored Jones polynomial
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Discrete Fourier transforms, quantum $6j$-symbols and deeply truncated tetrahedra
• Knot state asymptotics I, AJ Conjecture and abelian representations
• A relative version of the Turaev-Viro invariants and the volume of hyperbolic polyhedral 3-manifolds
• Rationality of the SL(2,C)-Reidemeister torsion in dimension 3
• Defect and degree of the Alexander polynomial
• Evolution properties of the knot's defect
• Density spectra for knots
• Turaev-Viro invariants, colored Jones polynomials and volume
• On the asymptotic behavior of the colored Jones polynomial of the figure-eight knot associated with a real number
• Volumes of Random Alternating Link Diagrams
• The colored Jones polynomial of the figure-eight knot and a quantum modularity
• Cyclotomic Expansion of Generalized Jones Polynomials
• On the asymptotic expansion for the relative Reshetikhin-Turaev invariants of fundamental shadow link pairs
• A Quantum Invariant of Links in $T^2\times I$ with Volume Conjecture Behavior
• Ihara zeta function and twisted Alexander invariants
• The Kauffman skein module at first order
• The Newton polygon of a recurrence sequence of polynomials and its role in TQFT