Kashaev–Murakami–Murakami volume conjecture for hyperbolic knots

Assume the Axiom of Choice and Countable Choice for the normalized colored Jones sequence used below.

Let $K$ be an oriented smooth knot in $S^3$, let $E_K$ be its compact exterior, and equip $\mathrm{int}(E_K)$ with a complete finite-volume hyperbolic metric $g$ having exactly one peripheral torus cusp. Write ${\mathrm{Vol}}_g(K)$ for its Riemannian volume. Let $J_N(K;q)\in \mathbb{Z}[q,q^{-1}]$ be the normalized $N$-dimensional colored Jones Laurent polynomial with $J_N(◯;q)=1$, in the variable $q=A^4$ declared in the generic color definition. Put $q_N=\exp (2\pi i\mathrm{/}N)$. The conjecture is that $J_N(K;q_N)\mathrm{\ne }0$ for all sufficiently large $N$ and that ${\lim }_{N\to \mathrm{\infty }}\frac{2\pi }{N}\log \mathrm{\mid }{J}_N(K;q_N)\mathrm{\mid }={\mathrm{Vol}}_g(K)\mathrm{.}$ This is the real absolute-value formulation; the complex-volume refinement is a separate conjecture.

Normalized colored Jones sequence

The normalized generic colored Jones quotient is an integral Laurent polynomial

Murakami and Murakami, The colored Jones polynomials and the simplicial volume of a knot, §§1–5

Murakami, An Introduction to the Volume Conjecture, §§1–4

As of 2026-10-19, the classical diagonal-sequence conjecture for arbitrary hyperbolic knots remains open in the inspected status source. Aistleitner and Borda (2025), §1, state that the classical volume conjecture is proved for some low-crossing knots, including the figure-eight knot, but remains open for general knots. Kakuta's version-3 preprint (2026), §1, proves a different varying-color, deformed-parameter statement for a range of cone angles on the figure-eight knot; its stated parameter and color regime do not settle the classical sequence with color dimension $N$ and parameter $q_N=\exp (2\pi i\mathrm{/}N)$ for every hyperbolic knot. The complex-volume/Chern–Simons refinement is distinct from the real limit stated above.

The local Laurent-polynomial supplier proves that the displayed generic normalization is an element of $\mathbb{Z}[q,q^{-1}]$ for every knot and color and matches Murakami’s explicit normalization with $q=A^4$. Thus $J_N(K;q_N)$ is evaluation of that Laurent polynomial. For $N\ge 2$ the quantum dimension in the generic quotient vanishes at $q_N$; direct specialized division is not its definition. This algebraic well-definedness does not prove the conjectured asymptotic limit.

Aistleitner and Borda, A conjecture of Zagier and the value distribution of quantum modular forms, §1, printed pp. 5044–5045

Kakuta, On the volume conjecture of the colored Jones invariants with arbitrary colors, version 3, §1

Habiro, On the quantum ${\mathfrak{s}\mathfrak{l}}_2$ invariants of knots and integral homology spheres, introduction and §§2.3, 3.3