On the volume conjecture of the colored Jones invariants with arbitrary colors

Abstract: We study the volume conjecture of the colored Jones invariants with sequences of colors corresponding to the deformation of the hyperbolic structure of a link complement. In particular, we investigate certain limits of the colored Jones invariants of the figure-eight knot and the Borromean rings and show that the limits are related to the volumes of hyperbolic cone manifolds whose singular sets are the links.

• The paper proves the volume conjecture by linking the exponential growth rate of colored Jones invariants to hyperbolic volumes through rigorous asymptotic analysis.
• Using detailed potential function techniques and the Poisson summation formula, the study establishes scaling limits for the figure-eight knot and Borromean rings.
• Extensive numerical experiments support the theoretical findings, extending the rigorous parameter range beyond previous results by Cho–Murakami and Wong–Yang.

Asymptotic Analysis of the Volume Conjecture for Colored Jones Invariants with Arbitrary Colors

Introduction and Background

The paper "On the volume conjecture of the colored Jones invariants with arbitrary colors" (2604.10472) addresses the asymptotic relationships between quantum link invariants and hyperbolic geometry, focusing on the colored Jones invariants evaluated in the setting where the coloring and the quantum parameters vary in a correlated fashion. The analysis targets a complexified version of the famous Volume Conjecture, originally posited by Kashaev and later extended by H. Murakami and J. Murakami. The core assertion is that the growth exponent of colored Jones invariants, when evaluated at roots of unity and under specific scaling limits, encodes geometric data: the hyperbolic volume of the link complement or of its hyperbolic cone manifold deformations.

The work systematically investigates this conjecture for two primary cases: the figure-eight knot and the Borromean rings. The results tie analytic asymptotics of the quantum invariants to the explicit hyperbolic volumes of cone manifolds, using detailed analysis of the associated potential functions.

Statement of the Conjecture and Main Results

Let $L$ be a hyperbolic link in $S^3$ with $\ell$ components. The colored Jones invariant $V_{j_1,\ldots,j_\ell}^{(r)}(L)$, colored by highest weights $j_i$ (one for each component), is considered at the root of unity $q = \exp(4\pi i/r)$. For a hyperbolic cone manifold $M_{\alpha_1,\ldots,\alpha_\ell}(L)$ with prescribed cone angles $\alpha_i$ around $L_i$, the Chen–Yang and Murakami conjecture predicts the following scaling:

$$\lim_{n\to\infty} \frac{4\pi}{2n+1} \log |V_{j_1,\ldots,j_\ell}^{(2n+1)}(L)| = \mathrm{Vol}(M_{\alpha_1,\ldots,\alpha_\ell}(L))$$

where the color parameters scale with $S^3$0 so that $S^3$1.

For the figure-eight knot ($S^3$2), Theorem 1 proves the conjecture for cone angles $S^3$3, where $S^3$4 is explicitly determined by the potential function and coincides with the largest parameter for which the relevant asymptotics dominate. For Borromean rings ($S^3$5), a multidimensional analogue holds for the set of cone angles where a corresponding criterion is satisfied. The criteria are expressed via the imaginary part of the potential function $S^3$6 associated to the link.

Notably, the proven range for cone angles surpasses those previously rigorously established by Cho–Murakami via colored Alexander invariants or by Wong–Yang for relative Reshetikhin–Turaev invariants.

Analytical Framework: Potential Function and Asymptotics

The analysis centers on expressing the colored Jones invariant for the link as a sum over $S^3$7 indexed by the color:

$S^3$8

where, in the large-$S^3$9 limit and under the prescribed scaling of $\ell$0, the summand is dominated by terms near stationary points of the phase function. This can be expressed asymptotically using the dilogarithm and the so-called potential function $\ell$1 (for $\ell$2) or $\ell$3 (for $\ell$4):

$\ell$5

and similarly for $\ell$6 with more complex dependence on all three cone angles.

The central analytic maneuver is to apply the Poisson summation formula, after localizing via smooth cut-off functions, to separate rapidly oscillating (alternating sign) regions of the summand from those of constant sign. Using the methods of Ohtsuki [Ohtsuki 2016], the exponential growth rate is shown to arise exclusively from stationary phase points where the imaginary part of $\ell$7 (or $\ell$8) is maximized, matching the known formulas for hyperbolic volumes via the Lobachevskii function and dilogarithms.

Additionally, the structure of the summation index is finely analyzed to partition the sum into alternating and non-alternating regimes, using precise trigonometric and quantum group considerations. The upper bound for the proven range of $\ell$9 (or the $V_{j_1,\ldots,j_\ell}^{(r)}(L)$0 region for $V_{j_1,\ldots,j_\ell}^{(r)}(L)$1) stems from the interplay between the maximum of the imaginary part of the potential function and the geometric volume formula.

Connection to Hyperbolic Geometry

The exponential growth rate of the colored Jones invariants, essentially the leading exponential in the asymptotic expansion, is shown to coincide with the hyperbolic volume of the deformed link complement (or cone manifold). For the figure-eight knot, the volume formula is given explicitly:

$V_{j_1,\ldots,j_\ell}^{(r)}(L)$2

where $V_{j_1,\ldots,j_\ell}^{(r)}(L)$3 is expressed in terms of $V_{j_1,\ldots,j_\ell}^{(r)}(L)$4.

A similar but more elaborate formula involving an extremal parameter $V_{j_1,\ldots,j_\ell}^{(r)}(L)$5 (the solution to a quartic in terms of the cone angles) gives the cone manifold volume for Borromean rings, matching known geometric constructions.

Numerical and Experimental Aspects

For larger cone angles outside the rigorously proven range, the paper supplements analytical results with extensive numerical experiments, observing that the conjectured asymptotic growth remains valid up to the hyperbolic structure's degeneration. These computations are consistent with established geometric transitions of cone manifolds.

Implications and Future Directions

The analysis robustly extends the class of link and color parameter regimes for which the complexified Volume Conjecture is rigorously verified. The results clarify the precise analytic–geometric correspondence through the explicit structure of the Jones polynomial's potential function, enhancing the link between quantum invariants and 3-manifold topology. The proof techniques, particularly the Poisson summation/analytic localization approach, are readily adaptable to more general links and quantum invariants, suggesting new avenues for extending such asymptotic identities.

The explicit linkage between quantum invariants at arbitrary colorings and geometric quantities under hyperbolic cone deformation deepens the analytic foundation for quantum topology and its interactions with hyperbolic geometry. This has implications for further study of quantum invariants of non-complete or singular geometric structures, the extension to other quantum group invariants, and potential applications to quantum Chern–Simons theory, building on work by Gukov and Murakami.

Conclusion

This work provides a comprehensive and rigorous asymptotic analysis of the colored Jones invariants for the figure-eight knot and Borromean rings, establishing the Chen–Yang/Murakami volume conjecture in a broader parameter space and employing advanced analytic methods to bridge quantum and geometric topology. The detailed potential function analysis, fine partitioning of summation indices, and integration with geometric formulas constitute substantial technical advancements, offering a framework and blueprint for similar results on more complex links and quantum invariants.