Parametrization of the volume conjecture for hyperbolic knots beyond the figure-eight knot

Murakami's parametrization of the volume conjecture (J. Topology, 2013; restated as Conjecture 1.6 of arXiv:2312.00350) predicts that for every hyperbolic knot $K$ the colored Jones polynomial $J_N(K;e^{(2\pi i+u)/N})$ has, for $u$ in a punctured neighborhood of $0$, a full leading asymptotic as $N\to\infty$ whose exponential growth is governed by the potential function $S_K(u)$ — recovering the $\mathrm{PSL}(2;\mathbb{C})$ Chern–Simons invariant of the deformed representation $\rho_u$ — and whose prefactor is the homological adjoint Reidemeister torsion $T_K(\rho_u)$. All rigorous verifications to date concern the figure-eight knot only: most recently for real $u>\operatorname{arccosh}(3/2)$ (arXiv:2312.00350, where $\rho_u$ is $\mathrm{SL}(2;\mathbb{R})$-valued) and for complex evaluation exponents $\xi$ with $0<\operatorname{Im}\xi<\pi/2$ (arXiv:2602.01049, Feb 2026). The problem is to prove or refute the conjecture for hyperbolic knots other than the figure-eight knot, up to full generality, with the sign convention of the adjoint Reidemeister torsion fixed — a point the source itself leaves unresolved even for the figure-eight knot.

Background

The colored Jones polynomial $J_N(K;q)$ of a knot $K\subset S^3$ is the sequence of quantum invariants obtained by evaluating the knot with the $N$-dimensional irreducible representation of the quantum group $U_q(\mathfrak{sl}_2)$; $J_2$ is the classical Jones polynomial and $q$ is a complex parameter. Kashaev (1997) introduced an invariant built from the quantum dilogarithm and conjectured that it grows exponentially in $N$ at a rate given by the hyperbolic volume of the knot complement; Murakami and Murakami (2001) identified Kashaev's invariant with $J_N(K;e^{2\pi i/N})$ and formulated the Volume Conjecture: for any knot $K$,

$$\lim_{N\to\infty}\frac{2\pi}{N}\log\left|J_N\left(K;e^{2\pi i/N}\right)\right|=\operatorname{Vol}(S^3\setminus K),$$

where $\operatorname{Vol}$ is the simplicial volume. A knot is hyperbolic when its complement carries a complete hyperbolic metric of finite volume; by Mostow rigidity this metric is unique, so the hyperbolic volume of the complement is a topological invariant, and for hyperbolic $K$ it coincides with the simplicial volume on the right-hand side. Dropping the absolute value gives a complex limit, conjectured by Murakami–Murakami–Okamoto–Takata–Yokota (2002) to equal the complex volume $\operatorname{Vol}(K)+i\,\mathrm{CS}^{\mathrm{SO}(3)}(K)$, where the Chern–Simons invariant is the secondary invariant of the Levi-Civita connection of the hyperbolic structure (the integral of the Chern–Simons 3-form over the complement, defined modulo $4\pi^2$). A finer refinement, developed by Gukov, by Dimofte–Gukov–Lenells–Zagier, and by Gukov–Murakami, predicts the subleading prefactor of the asymptotics in terms of the adjoint Reidemeister torsion $T_K(\rho)$ — the Whitehead torsion of the cochain complex of $S^3\setminus K$ with coefficients twisted by $\operatorname{Ad}\circ\rho$ for a flat $\mathrm{SL}(2;\mathbb{C})$ connection $\rho$, a classical secondary invariant measuring the homological size of the flat connection.

Murakami (Journal of Topology, 2013) proposed the parametrized version: one perturbs the evaluation point $2\pi i$ by a complex number $u$ and studies $J_N(K;e^{\xi/N})$, where $\xi:=u+2p\pi i$ is the exponent of the evaluation point at winding $p$, so that $u=\xi-2p\pi i$ is the deformation parameter; the unperturbed point is $p=1$, $u=0$, i.e. $\xi=2\pi i$, while the $p=0$ case is $\xi=u$. Let $(\mu,\lambda)$ be the standard meridian/longitude pair of $K$, and let $\rho_u:\pi_1(S^3\setminus K)\to\mathrm{SL}(2;\mathbb{C})$ be the irreducible representation — deforming the holonomy representation for small $|u|$ — under which the meridian has eigenvalues $e^{\pm u/2}$ and the longitude has eigenvalues $-e^{\pm v(u)/2}$ for a companion function $v(u)$. The conjecture predicts a complete leading asymptotic combining three classical ingredients: a potential function $S_K(u)$ (a Neumann–Zagier-type potential encoding the complex volume of the deformed structure), the homological adjoint Reidemeister torsion $T_K(\rho_u)$ in Porti's normalization with the meridian as homology basis, and the $\mathrm{PSL}(2;\mathbb{C})$ Chern–Simons invariant; the precise formula is stated in the problem statement. Physically this is the predicted semiclassical asymptotics of $\mathrm{SL}(2;\mathbb{C})$ complex Chern–Simons theory, of which the colored Jones polynomial (the $\mathrm{SU}(2)$ theory) at the perturbed coupling is expected to be a sector.

The conjecture is proved only for the figure-eight knot $4_1$, in the regimes examined so far: the logarithmic limit for small $|u|$ (Murakami–Yokota 2007); the full formula for small $|u|$ with nonzero winding $p$, i.e. evaluation at $\xi=u+2p\pi i$ (Murakami 2013); the boundary value $u=\kappa=\operatorname{arccosh}(3/2)$, where the representation degenerates to the parabolic one (Murakami, Algebraic & Geometric Topology 2025; polynomial growth at $p=0$ was known from Hikami–Murakami 2008); real $u>\kappa$ with $p=0$ (Murakami–Tran 2025) and with integer $p\ge 1$ (Murakami, arXiv:2312.00350, where $\rho_u$ is $\mathrm{SL}(2;\mathbb{R})$-valued and the formula acquires an additional multiplicative factor $J_p(\mathscr{E};e^{4N\pi^2/\xi})$); the unperturbed point $\xi=2\pi i$ (Andersen–Hansen 2006); and complex $\xi$ with $0<\operatorname{Im}\xi<\pi/2$ on the $p=0$ case of the parametrization, where $\xi=u$ (Murakami, arXiv:2602.01049). For hyperbolic knots other than $4_1$ only the single-point refinement at the unperturbed evaluation $\xi=2\pi i$ (torsion and Chern–Simons at that one point) is known, for knots with at most seven crossings (Andersen–Hansen for $4_1$; Ohtsuki for $5_2$ and the seven-crossing knots; Ohtsuki–Yokota for $6_1,6_2,6_3$), plus the ordinary volume conjecture (leading exponential rate only, no torsion or Chern–Simons prefactor) for some special families such as torus knots, twist knots, and double twist knots. Garoufalidis and Lê (2011) proved that for any knot $J_N(K;e^{u/N})\to 1/\Delta(K;e^u)$ for $u$ near $0$, where $\Delta$ is the Alexander polynomial, so the exponential growth concentrates on the $2\pi i$-sheets that the conjecture addresses. No verification of the parametrized conjecture — the full formula with torsion and Chern–Simons as functions of the deformation parameter $u$ — exists for any hyperbolic knot other than $4_1$.

Problem Statement

Prove or refute the parametrization of the volume conjecture for hyperbolic knots other than the figure-eight knot. The conjecture (Murakami, J. Topology 6 (2013), 193–216; restated as Conjecture 1.6 of arXiv:2312.00350) asserts: for a hyperbolic knot $K$, there exists a neighborhood $U$ of $0$ in $\mathbb{C}$ such that for $u\in U\setminus\pi i\mathbb{Q}$,

$$J_N\left(K;e^{(2\pi i+u)/N}\right)\ \underset{N\to\infty}{\sim}\ \frac{\sqrt{-\pi}}{2\sinh(u/2)}\,T_K(\rho_u)^{-1/2}\left(\frac{N}{2\pi i+u}\right)^{1/2}\exp\left(\frac{S_K(u)}{2\pi i+u}\,N\right),$$

where $\rho_u$ is the irreducible $\mathrm{SL}(2;\mathbb{C})$ representation of $\pi_1(S^3\setminus K)$ with meridian eigenvalues $e^{\pm u/2}$, deforming the holonomy representation for small $|u|$; $T_K(\rho_u)$ is the homological adjoint Reidemeister torsion in Porti's normalization (meridian as homology generator); $S_K(u)$ is the potential function, from which the Chern–Simons invariant is recovered as $\mathrm{CS}_{u,v(u)}(\rho_u)=S_K(u)-u\pi i-u\,v(u)/4$ with $v(u)=2\,\frac{d}{du}S_K(u)-2\pi i$.

Since the figure-eight knot case is settled in the regimes listed in the background, the open content is the conjecture for every hyperbolic knot other than the figure-eight knot, up to and including a proof for all hyperbolic knots; a proof for any single further knot or family would be partial progress rather than a full resolution. Any resolution must in particular fix the sign convention under which $T_K(\rho_u)$ enters the formula: the homological adjoint Reidemeister torsion is defined only up to an overall sign, and even the figure-eight identification in arXiv:2312.00350 is stated only 'up to a sign' (following Porti). A resolution by counterexample must exhibit a specific hyperbolic knot $K$ and admissible parameters $u_n\to0$ (with each $\rho_{u_n}$ irreducible and $\mu$-regular in the sense of Porti, that is, regular with respect to the meridian so that the meridian-based torsion $T_K(\rho_{u_n})$ is defined) along which the asymptotic formula provably fails; equivalently, it must show that no neighborhood $U$ of $0$ satisfies the conjecture. Because the conjecture asserts the existence of $U$, failure at admissible parameters that do not accumulate at $0$ is not a refutation: a smaller neighborhood can exclude them.

References

1. Hitoshi Murakami, The colored Jones polynomial of the figure-eight knot and an SL(2;R)-representation, arXiv:2312.00350 (2023). DOI: 10.48550/arXiv.2312.00350, https://arxiv.org/abs/2312.00350
2. Hitoshi Murakami, The coloured Jones polynomial, the Chern–Simons invariant, and the Reidemeister torsion of the figure-eight knot, Journal of Topology 6 (2013), no. 1, 193–216. DOI: 10.1112/jtopol/jts036, https://doi.org/10.1112/jtopol/jts036
3. Hitoshi Murakami, The generalized volume conjecture for the figure-eight knot parametrized by a complex number with small imaginary part, arXiv:2602.01049 (2026). DOI: 10.48550/arXiv.2602.01049, https://arxiv.org/abs/2602.01049
4. Hitoshi Murakami, The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation, Algebraic & Geometric Topology 25 (2025), 3523–3583. DOI: 10.2140/agt.2025.25.3523, https://doi.org/10.2140/agt.2025.25.3523
5. Hitoshi Murakami and Anh T. Tran, On the Asymptotic Behavior of the Colored Jones Polynomial of the Figure-Eight Knot Associated with a Real Number, in: Low Dimensional Topology and Number Theory, Springer Proceedings in Mathematics & Statistics (2025), 175–209. DOI: 10.1007/978-981-97-3778-9_6, https://doi.org/10.1007/978-981-97-3778-9_6
6. Hitoshi Murakami and Yoshiyuki Yokota, The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces, Journal für die reine und angewandte Mathematik 607 (2007), 47–68. DOI: 10.1515/crelle.2007.045, https://doi.org/10.1515/crelle.2007.045
7. Hitoshi Murakami and Jun Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Mathematica 186 (2001), no. 1, 85–104. DOI: 10.1007/BF02392716, https://doi.org/10.1007/BF02392716
8. Hitoshi Murakami, Jun Murakami, Miyuki Okamoto, Toshie Takata, and Yoshiyuki Yokota, Kashaev's Conjecture and the Chern–Simons Invariants of Knots and Links, Experimental Mathematics 11 (2002), no. 3, 427–435. DOI: 10.1080/10586458.2002.10504485, https://doi.org/10.1080/10586458.2002.10504485
9. R. M. Kashaev, The Hyperbolic Volume of Knots from the Quantum Dilogarithm, Letters in Mathematical Physics 39 (1997), no. 3, 269–275. DOI: 10.1023/A:1007364912784, https://doi.org/10.1023/A:1007364912784
10. Sergei Gukov and Hitoshi Murakami, SL(2,C) Chern–Simons theory and the asymptotic behavior of the colored Jones polynomial, in: Modular Forms and String Duality, Fields Institute Communications 54, American Mathematical Society, Providence, RI (2008), 261–277. DOI: 10.1090/fic/054/12, https://doi.org/10.1090/fic/054/12
11. Joan Porti, Torsion de Reidemeister pour les variétés hyperboliques, Memoirs of the American Mathematical Society 128 (1997), no. 612. DOI: 10.1090/memo/0612, https://doi.org/10.1090/memo/0612
12. Stavros Garoufalidis and Thang T. Q. Lê, Asymptotics of the colored Jones function of a knot, Geometry & Topology 15 (2011), no. 4, 2135–2180. DOI: 10.2140/gt.2011.15.2135, https://doi.org/10.2140/gt.2011.15.2135
13. Jørgen Ellegaard Andersen and Søren Kold Hansen, Asymptotics of the quantum invariants for surgeries on the figure 8 knot, Journal of Knot Theory and Its Ramifications 15 (2006), no. 4, 479–548. DOI: 10.1142/S0218216506004555, https://doi.org/10.1142/S0218216506004555
14. Tomotada Ohtsuki, On the asymptotic expansion of the Kashaev invariant of the 5_2 knot, Quantum Topology 7 (2016), no. 4, 669–735. DOI: 10.4171/qt/83, https://doi.org/10.4171/qt/83
15. Tomotada Ohtsuki, On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings, International Journal of Mathematics 28 (2017), no. 13, 1750096. DOI: 10.1142/S0129167X17500963, https://doi.org/10.1142/S0129167X17500963
16. Tomotada Ohtsuki and Yoshiyuki Yokota, On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings, Mathematical Proceedings of the Cambridge Philosophical Society 165 (2018), no. 2, 287–339. DOI: 10.1017/S0305004117000494, https://doi.org/10.1017/S0305004117000494
17. Kazuhiro Hikami and Hitoshi Murakami, Colored Jones polynomials with polynomial growth, Communications in Contemporary Mathematics 10 (2008), suppl. 1, 815–834. DOI: 10.1142/S0219199708003034, https://doi.org/10.1142/S0219199708003034
18. Jun Murakami, Complexified tetrahedrons, fundamental groups, and volume conjecture for double twist knots, arXiv:2501.00225 (2025). DOI: 10.48550/arXiv.2501.00225, https://arxiv.org/abs/2501.00225
19. Fathi Ben Aribi, Antonin Guilloux, and Ka Ho Wong, FAMED by computer: proving the Andersen–Kashaev volume conjecture for 42,000 knots, arXiv:2512.17437 (2025). DOI: 10.48550/arXiv.2512.17437, https://arxiv.org/abs/2512.17437
20. Shinichiro Kakuta, On the volume conjecture of the colored Jones invariants with arbitrary colors, arXiv:2604.10472 (2026). DOI: 10.48550/arXiv.2604.10472, https://arxiv.org/abs/2604.10472