One-parameter-family asymptotic expansions of colored Jones polynomials and Turaev–Viro invariants for general hyperbolic knots (Wong–Au Conjectures 3–5, and the figure-eight regime $s\sim 1/2$)

For the figure-eight knot, Wong and Au proved a uniform asymptotic expansion formula (AEF) for the $M$-th colored Jones polynomial evaluated at the $(N+1/2)$-th root of unity with limiting color ratio $s=M/(N+1/2)$ near $1$, showed that colors with $s$ near $1$ dominate the Turaev–Viro invariant, and deduced the full AEF of the Turaev–Viro invariant with exponential rate the hyperbolic volume and prefactor involving the twisted Reidemeister torsion. They formulated three conjectures extending this iterated saddle-point program to every hyperbolic knot — a one-parameter family of holomorphic potential functions recovering the complex volume (Conjecture 3), dominance of colors with $s$ near $1$ in the Turaev–Viro color sum (Conjecture 4), and a bi-parameter potential assembling the colored Jones AEFs into a Turaev–Viro AEF (Conjecture 5) — and acknowledged that for the figure-eight knot itself no explicit AEF was obtained for $s$ near $1/2$, only a strict subdominance bound. All three conjectures remain open for general hyperbolic knots; the partial progress to date concerns twist knots and Whitehead-link surgeries.

Background

The $M$-th colored Jones polynomial $J_M(K;q)$ is the quantum $sl_2$ invariant of $K$ colored by the $M$-dimensional irreducible representation, a sequence of polynomials in $q$ indexed by the color $M$. The volume conjecture of Kashaev, equivalently formulated by Murakami and Murakami, predicts that the $N$-th colored Jones polynomial of a hyperbolic knot $K$ evaluated at the $N$-th root of unity grows exponentially at rate the hyperbolic volume: $\lim_{N\to\infty}\frac{2\pi}{N}\log|J_N(K;e^{2\pi i/N})|=\mathrm{Vol}(S^3\setminus K)$. A refinement due to H. Murakami gives a full asymptotic expansion formula (AEF) for $J_N(4_1)$ near the root of unity, obtained by expressing the invariant through an integral of the quantum dilogarithm (a $q$-deformation of Euler's dilogarithm) and applying the saddle point (Laplace) method: the exponent is governed by a holomorphic potential function $\Phi(z)$, and the leading term is $\text{const}\times(N/2\pi i)^{3/2}\exp(N\Phi(z_0))/\sqrt{\Phi''(z_0)}$ where $z_0$ is a non-degenerate critical point of $\Phi$; the critical value recovers the complex volume $\frac{1}{2\pi}(\mathrm{Vol}(K)+i\,\mathrm{CS}(K))$, combining the hyperbolic volume with the Chern–Simons invariant, a secondary real-valued invariant of the hyperbolic structure.

The Turaev–Viro invariant $TV_r(M)$ of a 3-manifold $M$ is the state-sum invariant built from quantum $6j$-symbols — the recoupling coefficients of the quantum group $U_q(sl_2)$ — evaluated at a root of unity (here $r$ is odd, $r=2N+1$, and the root at which the colored Jones polynomials are evaluated is $e^{2\pi i/(N+1/2)}$). Chen and Yang conjectured, and in cases proved, a volume conjecture for $TV_r$: $\lim_{r\to\infty}\frac{2\pi}{r}\log TV_r(M,e^{2\pi i/r})=\mathrm{Vol}(M)$ for hyperbolic $M$, with $r$ running over odd integers. Detcherry, Kalfagianni, and Yang expressed the Turaev–Viro invariant of a link complement as a finite sum of squared colored Jones polynomials: for a link $L$ with $n$ components and odd $r=2N+1\ge3$,

$$TV_r\!\left(S^3\setminus L,e^{\frac{2\pi i}{r}}\right)=2^{n-1}\left(\eta_r'\right)^2\sum_{1\le M\le N}\left|\tilde J_M\!\left(L;e^{\frac{2\pi i}{N+1/2}}\right)\right|^2,\qquad\eta_r'=\frac{2\sin(2\pi/r)}{\sqrt r},$$

where $\tilde J_M(L;q)=\frac{q^{M/2}-q^{-M/2}}{q^{1/2}-q^{-1/2}}J_M(L;q)$ is the unnormalized colored Jones polynomial. They used this formula to prove the Chen–Yang conjecture for the figure-eight knot complement and the Borromean rings complement, by showing that colors near the diagonal dominate the sum.

Wong and Au made this dominance argument effective for the figure-eight knot $4_1$ and upgraded it to full asymptotic expansions. They studied $J_M(4_1;e^{2\pi i/(N+1/2)})$ with the two colors linked by the limiting ratio $s=\lim_{N\to\infty}M/(N+1/2)$: they proved a uniform AEF for $s$ in a neighborhood of $1$ (the parameter $s$ is interpreted as a parametrization of points on the character variety of the knot complement — the algebraic set of $SL_2(\mathbb C)$-representations of the knot group modulo conjugation — with $s=1$ corresponding to the complete hyperbolic structure); they proved that for $s$ near $1/2$ the exponential growth rate is strictly smaller than the hyperbolic volume, so those colors are subdominant; and, combining these, they obtained the full AEF of $TV_r(S^3\setminus 4_1)$, whose leading exponential rate is $r\,\mathrm{Vol}(4_1)/(2\pi)$ and whose prefactor is $\frac{1}{4}(r/2)^{1/2}|T(K)|^{3/2}$, with $T(K)$ the twisted Reidemeister torsion of the unique complete hyperbolic structure (computed in the adjoint representation in the refinement literature; $T(4_1)=2/\sqrt{-3}$), the standard secondary invariant conjectured by Witten to govern the leading coefficient of quantum invariants. Their method breaks the naive two-dimensional saddle point approximation into iterated one-dimensional ones: first the AEF of each colored Jones polynomial in the color ratio $s$, then a saddle point approximation again along $s$.

The paper formulates, but does not prove, the extension of this entire program from $4_1$ to an arbitrary hyperbolic knot $K$, as Conjectures 3, 4, and 5 stated below, and records one concrete limitation even for $4_1$: no explicit full AEF for $J_M(4_1)$ in the regime $s\sim1/2$ was found, only an upper bound giving strict subdominance.

Problem Statement

Extend the iterated saddle-point asymptotic expansion program proved by Wong and Au for the figure-eight knot to arbitrary hyperbolic knots, as asked by Conjectures 3–5 of arXiv:1711.11290 (published as Algebraic & Geometric Topology 22 (2022)), together with the figure-eight completion the same authors explicitly left open. Throughout, $K$ is a hyperbolic knot in $S^3$; $M$ and $N$ are positive integers with $M\le N$; the evaluation point is the $(N+1/2)$-th root of unity $e^{2\pi i/(N+1/2)}$; and $s=\lim_{N\to\infty}M/(N+1/2)\in\mathbb R$ is the limiting ratio of the color $M$ to the root index. Here $J_M$ is the $M$-th colored Jones polynomial, $\tilde J_M(L;q)=\frac{q^{M/2}-q^{-M/2}}{q^{1/2}-q^{-1/2}}J_M(L;q)$ is the unnormalized colored Jones polynomial, and $\mathrm{Vol}(K)$, $\mathrm{CS}(K)$ denote the hyperbolic volume and Chern–Simons invariant of $S^3\setminus K$.

(1) One-parameter family of potential functions (Conjecture 3). There exists a small neighborhood $U\subset\mathbb R$ of $s=1$ such that for any $M,N$ with $s\in U$ there is a holomorphic potential function $\Phi_M^{(s)}(z)$ with the following properties.

(i) In the diagonal case $s=1$, the potentials converge to the classical potential of $K$:

$$\lim_{N\to\infty}\Phi_N^{(1)}(z)=\Phi^{(1)}_\infty(z).$$

(ii) Let $z^{(1)}_\infty$ be a non-degenerate saddle point of $\Phi^{(1)}_\infty$ giving the complex volume of $K$, i.e. $\frac{d}{dz}\Phi^{(1)}_\infty(z^{(1)}_\infty)=0$ and $\frac{d^2}{dz^2}\Phi^{(1)}_\infty(z^{(1)}_\infty)\ne0$, with critical value $\Phi^{(1)}_\infty(z^{(1)}_\infty)=\frac{1}{2\pi}[\mathrm{Vol}(K)+i\,\mathrm{CS}(K)]$. Then there is a smooth choice of non-degenerate saddle points $z_M^{(s)}$ of the family, i.e. $\frac{d}{dz}\Phi_M^{(s)}(z_M^{(s)})=0$ and $\frac{d^2}{dz^2}\Phi_M^{(s)}(z_M^{(s)})\ne0$, such that as $M\to N$ (equivalently, as $s\to1$),

$$z_M^{(s)}\to z^{(1)}_\infty \quad\text{and}\quad \Phi_M^{(s)}(z_M^{(s)})\to\Phi^{(1)}_\infty(z^{(1)}_\infty)=\frac{1}{2\pi}[\mathrm{Vol}(K)+i\,\mathrm{CS}(K)].$$

(iii) The family of potential functions determines the AEF of the colored Jones polynomial:

$$J_M\!\left(K,e^{\frac{2\pi i}{N+1/2}}\right)\sim_{M\to\infty}(\text{constant})\times\left(\frac{N}{2\pi i}\right)^{3/2}\frac{\exp\big((N+1/2)\,\Phi_M^{(s)}(z_M^{(s)})\big)}{\sqrt{\frac{d^2}{dz^2}\Phi_M^{(s)}(z_M^{(s)})}}.$$

(2) Dominance (Conjecture 4). With $U$ as in (1), the colors with $s\in U$ dominate the complementary colors in the Turaev–Viro color sum:

$$\sum_{M:\,s\notin U}\Big|\tilde J_M\!\left(K,e^{\frac{2\pi i}{N+1/2}}\right)\Big|^2=o\!\left(\sum_{M:\,s\in U}\Big|\tilde J_M\!\left(K,e^{\frac{2\pi i}{N+1/2}}\right)\Big|^2\right),$$

that is, the ratio of the left-hand sum to the right-hand sum tends to $0$ as $N\to\infty$.

(3) Bi-parameter assembly into the Turaev–Viro AEF (Conjecture 5). There exists a function $\Phi(s,z):D\subset U\times\mathbb C\to\mathbb C$, holomorphic in $z$, that recovers the potentials of (1) via $\Phi\big(\frac{M}{N+1/2},z\big)=\Phi_M^{(s)}(z)$ and $\Phi(1,z)=\Phi^{(1)}_\infty(z)$, together with a smooth choice of non-degenerate saddle points $z(s)$ for each $s\in D$ ($\frac{d}{dz}\Phi(s,z(s))=0$, $\frac{d^2}{dz^2}\Phi(s,z(s))\ne0$), such that the saddle point (Laplace-type) approximation applied again along the color parameter $s$ yields the AEF of the Turaev–Viro invariant at $r=2N+1$:

$$TV_r(K)=TV_r\!\left(S^3\setminus K,e^{\frac{2\pi i}{r}}\right)=\left(\eta_r'\right)^2\sum_{1\le M\le N}\Big|\tilde J_M\!\left(K,e^{\frac{2\pi i}{N+1/2}}\right)\Big|^2\sim_{N\to\infty}\frac{1}{4}\left(\frac{r}{2}\right)^{1/2}\big|T(K)\big|^{3/2}\exp\!\left(\frac{r}{2\pi}\,\mathrm{Vol}(S^3\setminus K)\right),$$

where $\eta_r'=2\sin(2\pi/r)/\sqrt r$ and $T(K)$ is the twisted Reidemeister torsion associated with the unique complete hyperbolic structure of $S^3\setminus K$ (for $K=4_1$, $T(4_1)=2/\sqrt{-3}$, and the display reduces to the published Theorem 8 of Wong–Au).

(4) Figure-eight knot at $s\sim\frac12$. Produce the explicit full asymptotic expansion formula for $J_M(4_1;e^{2\pi i/(N+1/2)})$ in the regime where the limiting ratio $s$ is near $\frac12$. What is currently known is only a strict subdominance bound (Lemma 2 of the source): there exist $\rho>0$ and a constant $k\in(0,1)$ independent of $s$ such that whenever $|s-\frac12|<\rho$,

$$\lim_{N\to\infty}\frac{2\pi}{N+1/2}\log\Big|J_M\!\left(4_1;e^{\frac{2\pi i}{N+1/2}}\right)\Big|\le k\,\mathrm{Vol}(S^3\setminus 4_1).$$

All four items (1)–(4) are required: items (1)–(3) are to be established for every hyperbolic knot $K$, and item (4) is required for $K=4_1$. Items (1)–(3) form one coherent objective, since the source states Conjecture 5 in terms of Conjectures 3 and 4, so that the bi-parameter assembly of (3) consumes the potential family of (1) and the dominance of (2). A proof restricted to a sub-family of knots, or covering only some of (1)–(4), does not resolve the task. The source notes that Conjecture 3 is true for $K=4_1$.

References

1. Ka Ho Wong and Thomas Kwok-Keung Au, "Asymptotic behavior of the colored Jones polynomials and Turaev–Viro invariants of the figure eight knot", Algebraic & Geometric Topology 22 (2022) 1–53, DOI 10.2140/agt.2022.22.1, https://arxiv.org/abs/1711.11290
2. Qingtao Chen and Tian Yang, "Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants", Quantum Topology 9 (2018) 419–460, DOI 10.4171/qt/111, https://arxiv.org/abs/1503.02547
3. Renaud Detcherry, Efstratia Kalfagianni, and Tian Yang, "Turaev–Viro invariants, colored Jones polynomials, and volume", Quantum Topology 9 (2018) 775–813, DOI 10.4171/qt/120, https://arxiv.org/abs/1701.07818
4. Rinat Kashaev, "The Hyperbolic Volume of Knots from the Quantum Dilogarithm", Letters in Mathematical Physics 39 (1997) 269–275, DOI 10.1023/A:1007364912784, https://link.springer.com/article/10.1023/A:1007364912784
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