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The generalized volume conjecture near the complete hyperbolic structure

Let H\mathscr{H} be a hyperbolic knot. Let ρu ⁣:π1(S3∖H)→SL⁡(2;C)\rho_u\colon\pi_1(S^3\setminus\mathscr{H})\to\operatorname{SL}(2;\mathbb{C}) be an irreducible representation that is a small deformation of the holonomy representation ρ0\rho_0. Let Tu(H)T_u(\mathscr{H}) be the corresponding cohomological adjoint Reidemeister torsion, let v(u)v(u) be determined by the meridian and preferred-longitude holonomies, and define

Su(H)=CS⁡u,v(u)(ρu)+uπ−1+uv(u)4.S_u(\mathscr{H})=\operatorname{CS}_{u,v(u)}(\rho_u)+u\pi\sqrt{-1}+\frac{uv(u)}{4}.

Generalized volume conjecture. For a hyperbolic knot H\mathscr{H}, there exists a neighbourhood U∈CU\in\mathbb{C} of 00 such that, if u∈U∖π−1Qu\in U\setminus\pi\sqrt{-1}\mathbb{Q}, then

JN(H;e(u+2π−1)/N)∼N→∞−π2sinh⁡(u/2)Tu(H)1/2(Nu+2π−1)1/2exp⁡(Su(H)u+2π−1N).J_N(\mathscr{H};e^{(u+2\pi\sqrt{-1})/N})\underset{N\to\infty}{\sim}\frac{\sqrt{-\pi}}{2\sinh(u/2)}T_u(\mathscr{H})^{1/2}\left(\frac{N}{u+2\pi\sqrt{-1}}\right)^{1/2}\exp\left(\frac{S_u(\mathscr{H})}{u+2\pi\sqrt{-1}}N\right).

This extends the refined volume conjecture by replacing the root-of-unity parameter with a nearby complex parameter and incorporating deformed Chern–Simons data and torsion. The source presents it as a conjecture proposed in the cited literature; its general status is open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Generalized Volume Conjecture near the complete hyperbolic structure

    Let HH be a hyperbolic knot and let ξ≠2π−1\xi\ne 2\pi\sqrt{-1} be a complex number close to 2π−12\pi\sqrt{-1}. Let CC, dd, S(ξ)S(\xi), and τ(ξ)\tau(\xi) be the quantities appearing in the asymptotic formula, where dd is a constant, and S(ξ)S(\xi) and τ(ξ)\tau(\xi) are respectively related to the SL⁡(2;C)\operatorname{SL}(2;\mathbb{C}) Chern--Simons invariant and the adjoint Reidemeister torsion associated with a representation of π1(S3∖H)\pi_1(S^3\setminus H) to SL⁡(2;C)\operatorname{SL}(2;\mathbb{C}). Generalized Volume Conjecture.

    JN(H;eξ/N)∼N→∞Csinh⁡(ξ)×τ(ξ)1/2(Nξ)dexp⁡(NξS(ξ)).J_N\left(H;e^{\xi/N}\right)\underset{N\to\infty}{\sim}\frac{C}{\sinh(\xi)}\times\tau(\xi)^{1/2}\left(\frac{N}{\xi}\right)^d\exp\left(\frac{N}{\xi}S(\xi)\right).

    This extends the complexified asymptotic behavior away from 2π−12\pi\sqrt{-1}; the source supplies no resolution status.

    source: Hitoshi Murakami and Anh T. Tran, “On the asymptotic behavior of the colored Jones polynomial of the figure-eight knot associated with a real number”, arXiv:2109.04664 (2022).

Sources & referencesView supporting material

Primary source

Hitoshi Murakami, “The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation”, arXiv:2307.07100 (2023).

colored-jones-polynomialcolored-jones-polynomialshyperbolic-knotsknot-theoryquantum-topologyvolume-conjecture

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