Volume Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $K\subset S^3$ be a tame knot, and let $J_N(K;q)$ be its $N$-colored Jones polynomial, defined using the $N$-dimensional irreducible $\mathfrak{sl}_2$ representation, with zero framing and normalization $J_N(\text{unknot};q)=1$. Use the convention $q=e^{2\pi i/N}$; in a convention using $s=q^{1/2}$, this is $s=e^{\pi i/N}$. For a hyperbolic knot, $\operatorname{Vol}(S^3\setminus K)$ is the volume of the complete finite-volume hyperbolic metric on its complement. More generally, cut the compact knot exterior along its canonical essential tori, and let $V(K)$ be the sum of the finite hyperbolic volumes of the hyperbolic pieces (zero if there are none). Equivalently $V(K)=v_3\|K\|$, where $v_3$ is the volume of a regular ideal hyperbolic tetrahedron and $\|K\|$ is the simplicial volume of the exterior. In particular $V(K)=\operatorname{Vol}(S^3\setminus K)$ for hyperbolic knots.

Statement recorded in the supplied source.

For every hyperbolic knot $K$, is \[ \lim_{N\to\infty}\frac{2\pi}{N} \log\left|J_N(K;e^{2\pi i/N})\right| =\operatorname{Vol}(S^3\setminus K)? \] The standard all-knot extension proposed in the cited source asks, for every knot $K$, whether the same limit exists and equals $V(K)$. When a Jones value is zero, its logarithm is interpreted as $-\infty$. [S2]

Short English statement

Does the exponential growth of normalized colored Jones values at successive roots of unity recover hyperbolic knot-complement volume? The all-knot extension asks for the sum of the hyperbolic volumes of the exterior's pieces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1485 (TOP-007). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hitoshi Murakami and Jun Murakami, The Colored Jones Polynomials and the Simplicial Volume of a Knot, Acta Mathematica 186 (2001), 85--104; arXiv:math/9905075v2, introduction, normalization before evaluation at the root of unity, and Section 5, Conjecture 5.1. https://arxiv.org/abs/math/9905075.