Math 2.0

Manifolds and cell complexes Three-manifolds Conjecture

Volume conjecture

Builds on KnotHyperbolic knotJones polynomialHyperbolic 3-manifold

Statement

Let KK be a hyperbolic knot in S3S^3, so that its complement S3∖KS^3 \setminus K is a hyperbolic 3-manifold of finite volume Vol(S3∖K)\mathrm{Vol}(S^3 \setminus K). For N≥2N \ge 2, let JN(K;q)J_N(K; q) be the NN-th colored Jones polynomial of KK, normalised so that its value on the unknot is 11. The conjecture states that lim⁡N→∞2πNlog⁡∣JN(K;e2πi/N)∣=Vol(S3∖K).\lim_{N \to \infty} \frac{2\pi}{N} \log \abs{J_N\left(K; e^{2\pi i / N}\right)} = \mathrm{Vol}(S^3 \setminus K).

Remarks

Kashaev formulated the conjecture in 1997 for invariants he had built from the quantum dilogarithm. H. Murakami and J. Murakami showed that Kashaev's invariant equals JN(K;e2πi/N)J_N(K; e^{2\pi i / N}). They extended the conjecture to all knots, with the simplicial volume of the complement in place of the hyperbolic volume. The conjecture is open in general. It has been verified for the figure-eight knot and for a few other hyperbolic knots. It would imply that a knot whose colored Jones polynomials all agree with those of the unknot has a complement of simplicial volume zero.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Hyperbolic 3-manifold, concept, no evidence levelHyperbolic 3-manifoldconceptHyperbolic knot, concept, no evidence levelHyperbolic knotconceptJones polynomial, concept, no evidence levelJones polynomialconceptKnot, concept, no evidence levelKnotconceptVolume conjecture, conjecture, no evidence levelVolume conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Hyperbolic 3-manifold concept
  2. Hyperbolic knot concept
  3. Jones polynomial concept
  4. Knot concept
  5. Volume conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Volume_conjecture
About this page
TagZP4AWD45CE
URIm2://ZP4AWD45CE
Kindconjecture
WritingW0 stub
MSC57K16, 57K32, 57K14
StatusThis copy of the site has no build data. Run m2 status ZP4AWD45CE.
AboutJones polynomial, Hyperbolic knot
UsesKnot, Hyperbolic knot, Jones polynomial, Hyperbolic 3-manifold
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Changed2 November 2026

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