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  "body": "\n## Statement\n\nLet $K$ be a [[hyperbolic knot|K47G2Q24DT]] in $S^3$, so that its complement $S^3 \\setminus K$ is a [[hyperbolic 3-manifold|GHVW1Z35BS]] of finite volume $\\mathrm{Vol}(S^3 \\setminus K)$.\nFor $N \\ge 2$, let $J_N(K; q)$ be the $N$-th colored [[Jones polynomial|4SPC17FPXE]] of $K$, normalised so that its value on the unknot is $1$.\nThe conjecture states that\n$$\\lim_{N \\to \\infty} \\frac{2\\pi}{N} \\log \\abs{J_N\\left(K; e^{2\\pi i / N}\\right)} = \\mathrm{Vol}(S^3 \\setminus K).$$\n\n## Remarks\n\nKashaev formulated the conjecture in 1997 for invariants he had built from the quantum dilogarithm.\nH. Murakami and J. Murakami showed that Kashaev's invariant equals $J_N(K; e^{2\\pi i / N})$.\nThey extended the conjecture to all knots, with the simplicial volume of the complement in place of the hyperbolic volume.\nThe conjecture is open in general.\nIt has been verified for the figure-eight knot and for a few other hyperbolic knots.\nIt would imply that a knot whose colored Jones polynomials all agree with those of the unknot has a complement of simplicial volume zero.\n\n## Explanations\n\n### L1 human\n\nThe colored Jones polynomials of a knot come from algebra and quantum physics.\nThe volume of the knot complement comes from hyperbolic geometry.\nThe conjecture says that the growth rate of the first, evaluated at special roots of unity, is exactly the second.\nIt would be a deep bridge between quantum invariants and geometry.\n",
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      "first_stated": "1997",
      "posed_by": "Rinat Kashaev"
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  "msc": [
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