Open problem, Topology, posed 1997

Volume conjecture (knots)

Open

For every knot $K \subset S^3$ , let $J_N(K; q)$ denote its $N$ -colored Jones polynomial normalized so that $J_N(\text{unknot}; q) = 1$ . Then $\lim_{N \to \infty} \frac{2\pi}{N} \log \left| J_N\!\left(K; e^{2\pi i / N}\right) \right| = \operatorname{Vol}(S^3 \setminus K) = v_3 \|S^3 \setminus K\|$ , where $v_3 \approx 1.0149416$ is the volume of a regular ideal hyperbolic tetrahedron and $\|S^3 \setminus K\|$ is the Gromov simplicial volume of the knot complement (equal to the hyperbolic volume when $K$ is a hyperbolic knot).

Research frontier as of 2026

As of 2026, the Kashaev–Murakami–Murakami volume conjecture remains open in general, though it has been proved for several explicit families of knots and links. Rigorous proofs exist for the figure-eight knot $4_1$ , torus knots, hyperbolic knots with at most $7$ crossings (Ohtsuki, 2016–2018), Whitehead doubles of non-trivial torus knots (Zheng, 2007), the Borromean rings, and specific families of twist knots and fundamental shadow links. For general hyperbolic knots, the main difficulty lies in controlling oscillatory state sums with many complex variables so that the saddle-point method can be applied without spurious critical points dominating the integral.

Best known results

Proved rigorously for the figure-eight knot $4_1$ , all torus knots, all hyperbolic knots with $\le 7$ crossings (Ohtsuki), Whitehead doubles of $(2,p)$ torus knots (Zheng), and fundamental shadow links.
The Chen–Yang volume conjecture for Turaev–Viro invariants at $q = e^{2\pi i/r}$ ( $r$ odd) has been proved for fundamental shadow links, octahedral links, and Dehn fillings with sufficiently large coefficients (Belletti, Detcherry, Kalfagianni, Yang).

Tools and where they stop

Tool	Achieved	Where it stops
Poisson summation and multidimensional saddle-point method (Yokota, Andersen–Hansen, Ohtsuki)	Replaces the state sum for $J_N(K; e^{2\pi i/N})$ with contour integrals of Faddeev's quantum dilogarithm whose critical points satisfy Thurston's hyperbolicity (gluing) equations, proving the conjecture and full asymptotic expansion for knots up to $7$ crossings	For diagrams with more crossings, deforming the integration contour to pass purely through the geometric saddle point without crossing poles or higher-real-part critical points requires case-by-case convex-hull and Hessian analyses
Quantum $A$ -polynomial and the AJ conjecture (Garoufalidis, Lê)	Shows that the sequence $J_N(K; q)$ is $q$ -holonomic and links its recurrence operator in the $q \to 1$ limit to the classical $\operatorname{SL}(2,\mathbb{C})$ character variety and the $A$ -polynomial of $K$	Determines the set of potential exponential growth rates via Mahler-measure/volume regulators on the character variety, but does not by itself identify which branch dominates at $q = e^{2\pi i/N}$

Open questions

Does the volume conjecture hold for all hyperbolic knots and links in $S^3$ ?
For satellite knots whose Gromov simplicial volume is strictly positive, does $\frac{2\pi}{N}\log|J_N(K; e^{2\pi i/N})|$ always converge to $v_3 \|S^3 \setminus K\|$ ?

References

Rinat M. Kashaev (1997). The hyperbolic volume of knots from the quantum dilogarithm · DOI:10.1023/A:1007364912784 · arXiv:q-alg/9601025v1
Hitoshi Murakami, Jun Murakami (2001). The colored Jones polynomials and the simplicial volume of a knot · DOI:10.1007/BF02392716 · arXiv:math/9905075v2
Hitoshi Murakami, Yoshiyuki Yokota (2018). An Introduction to the Volume Conjecture · DOI:10.1007/978-981-13-1150-5