Math Conjectures

Hyperbolic volume conjecture

OPENMajorConjectureProposed 1997 · Canonical special case

Canonical statement

Let K⊂S3K\subset S^{3} be a hyperbolic knot and let JN(K;q)J_N(K;q) be its NN-colored Jones polynomial normalized by JN(unknot;q)=1J_N(\text{unknot};q)=1. Then
lim⁡N→∞2πNlog⁡∣JN ⁣(K;e2πi/N)∣=Vol⁡(S3∖K), \lim_{N\to\infty}\frac{2\pi}{N} \log\left|J_N\!\left(K;e^{2\pi i/N}\right)\right| =\operatorname{Vol}(S^{3}\setminus K),
where the right-hand side is the complete finite-volume hyperbolic volume.
View source LaTeX
Let \(K\subset S^{3}\) be a hyperbolic knot and let \(J_N(K;q)\) be its \(N\)-colored Jones polynomial normalized by \(J_N(\text{unknot};q)=1\). Then \[ \lim_{N\to\infty}\frac{2\pi}{N} \log\left|J_N\!\left(K;e^{2\pi i/N}\right)\right| =\operatorname{Vol}(S^{3}\setminus K), \] where the right-hand side is the complete finite-volume hyperbolic volume.
GEO-TOP-020:[Kashaev1997HyperbolicVolume][MurakamiMurakami2001ColoredJones][GaroufalidisLe2005Asymptotics][GaroufalidisEtAl2025FAMED]
Geometry and topologyMathematical physics

Notes

The volume conjecture links quantum invariants to hyperbolic geometry. Kashaev observed in 1997 that an invariant he had constructed from the quantum dilogarithm appeared to grow exponentially at a rate given by the hyperbolic volume of the knot complement [Kashaev1997HyperbolicVolume]. Murakami and Murakami then identified Kashaev's invariant with the NN-colored Jones polynomial evaluated at the root of unity q=e2πi/Nq=e^{2\pi i/N}, putting the conjecture in its standard form: for a hyperbolic knot KK, the quantity 2πNlog⁡∣JN(K;e2πi/N)∣\frac{2\pi}{N}\log|J_N(K;e^{2\pi i/N})| should converge to Vol⁡(S3∖K)\operatorname{Vol}(S^3\setminus K) as N→∞N\to\infty [MurakamiMurakami2001ColoredJones].

The conjecture is proved for the figure-eight knot and for several substantial families, and the general asymptotics of the colored Jones function have been analyzed in depth by Garoufalidis and Lê [GaroufalidisLe2005Asymptotics]. More recently, computer-assisted methods established the related Andersen–Kashaev version of the conjecture for some 42,000 knots [GaroufalidisEtAl2025FAMED].

For a general hyperbolic knot both the existence of the limit and the equality with the volume remain unproven; the stronger complex-volume and all-link versions, not treated as separate entries here, remain open as well.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.