Let mathscrH⊂ S^3 be a hyperbolic knot. Let operatornamecv(mathscrH) be its complex volume, and let T(mathscrH) be the adjoint cohomological Reidemeister torsion twisted by the holonomy representation rho_0 colonpi_1(S^3 setminusmathscrH)→operatornameSL(2;C). For functions F and G, write F(N)undersetN→∞simG(N) when lim_N→∞F(N)/G(N)=1. Refined volume conjecture. Let mathscrH⊂ S^3 be a hyperbolic knot. Then J_N(mathscrH;e^2 pi√-1/N)undersetN→∞sim(fracT(mathscrH)2√-1)^1/2N^3/2exp(fracoperatornamecv(mathscrH)2 pi√-1N). The conjecture refines the complexified volume conjecture by predicting the power-law prefactor and Reidemeister-torsion contribution. The source states that it has been proved for the figure-eight knot and for hyperbolic knots with at most seven crossings.

The refined volume conjecture for hyperbolic knots

Let $\mathscr{H}\subset S^3$ be a hyperbolic knot. Let $\operatorname{cv}(\mathscr{H})$ be its complex volume, and let $T(\mathscr{H})$ be the adjoint cohomological Reidemeister torsion twisted by the holonomy representation

\rho_0\colon\pi_1(S^3\setminus\mathscr{H})\to\operatorname{SL}(2;\mathbb{C}).

For functions $F$ and $G$ , write $F(N)\underset{N\to\infty}{\sim}G(N)$ when $\lim_{N\to\infty}F(N)/G(N)=1$ . Refined volume conjecture. Let $\mathscr{H}\subset S^3$ be a hyperbolic knot. Then

J_N(\mathscr{H};e^{2\pi\sqrt{-1}/N})\underset{N\to\infty}{\sim}\left(\frac{T(\mathscr{H})}{2\sqrt{-1}}\right)^{1/2}N^{3/2}\exp\left(\frac{\operatorname{cv}(\mathscr{H})}{2\pi\sqrt{-1}}N\right).

The conjecture refines the complexified volume conjecture by predicting the power-law prefactor and Reidemeister-torsion contribution. The source states that it has been proved for the figure-eight knot and for hyperbolic knots with at most seven crossings.

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.

The refined volume conjecture for hyperbolic knots
Let $K$ be a hyperbolic knot. Let $\operatorname{Vol}(K)$ be its hyperbolic volume, let $\operatorname{CS}^{\mathrm{SO}(3)}(K)$ be its $\mathrm{SO}(3)$ Chern--Simons invariant, and let $\tau(K)$ be determined by the condition that $2\sqrt{-1}\tau(K)^{-2}$ equals the homological adjoint Reidemeister torsion twisted by the holonomy representation associated with the complete hyperbolic structure. Refinement of Kashaev's conjecture.
$J_N\left(K;e^{2\pi\sqrt{-1}/N}\right)\underset{N\to\infty}{\sim}\tau(K)N^{3/2}\exp\left(\frac{N}{2\pi}\bigl(\operatorname{Vol}(K)+\sqrt{-1}\operatorname{CS}^{\mathrm{SO}(3)}(K)\bigr)\right).$
The statement refines the complexified volume conjecture by specifying the leading polynomial factor and torsion. It is proved in the source for hyperbolic knots with at most seven crossings, but remains open in general.
source: Hitoshi Murakami, “The colored Jones polynomial of the figure-eight knot and an SL(2;R)-representation”, arXiv:2312.00350 (2023).

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).

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The refined volume conjecture for hyperbolic knots

Equivalent formulations 1

The refined volume conjecture for hyperbolic knots

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