Chen-Yang Volume Conjecture for Turaev-Viro Invariants

Chen-Yang Volume Conjecture for Turaev-Viro Invariants

Definitions and mathematical statement

Let $M$ be a finite-volume hyperbolic three-manifold in the closed, cusped, or totally geodesic boundary class specified in Chen--Yang's Conjecture 1.1. Its hyperbolic volume $\operatorname{Vol}(M)$ uses sectional curvature $-1$. For odd integers $r$, $\operatorname{TV}_r(M;q)$ is the Turaev--Viro quantum invariant: a finite sum over admissible labels of a triangulation using quantum dimensions and quantum $6j$-symbols. For cusps and boundary, use precisely the compact-core extension and normalization in the source.

The assertion is \[ \lim_{\substack{r\to\infty\\r\text{ odd}}} \frac{2\pi}{r}\log\operatorname{TV}_r(M;e^{2\pi i/r}) =\operatorname{Vol}(M). \] The choice of root of unity is part of the problem. The logarithm and invariant normalization are those of the stated source; changing roots or replacing this invariant by a different quantum invariant changes the target. This is an asymptotic growth statement, not equality at finite level.

Formulation and concept references: [S1].

Short English statement

Does the exponential growth rate of these Turaev--Viro invariants recover hyperbolic volume for every manifold in the stated class?

Research attempt

Scope, attribution, and source check.

This unresolved attempt by GPT-6 Astra Ultra was prepared on 27 September 2026 using the Chen--Yang primary source [S1], exact normalization checks and elementary asymptotic estimates. The root $e^{2\pi i/r}$, odd levels, curvature $-1$ volume and the source's boundary conventions are retained. The positive results for specified link complements in [E1] do not settle all manifolds in the catalogue. The attempt analyzes a quantum-factorial building block and formulates the precise lower-bound problem that remains after estimating individual summands.

Polynomial factors do not affect the exponential rate.

Fix a triangulation with $E$ edges. At level $r$ the number of edge labelings is at most $r^E$, and admissibility only reduces this count. If a positive normalization factor $p_r$ satisfies $r^{-C}\leq p_r\leq r^C$, then \[ \frac{2\pi}{r}\log p_r\longrightarrow0. \] Thus polynomial counting and normalization factors can be controlled without knowing the exact leading coefficient of the invariant.

This observation does not authorize arbitrary renormalization. A factor $e^{cr}$ changes the limiting rate by $2\pi c$. Nor can one replace the root of unity by a different one: it changes the quantum-number factors themselves and can change their exponential behavior. All estimates below keep the source's root fixed.

An elementary quantum-factorial asymptotic.

For odd $r$ and $0\leq k\leq(r-1)/2$, put \[ F_r(k)=\prod_{j=1}^k2\sin(2\pi j/r),\qquad F_r(0)=1, \qquad \Lambda(\theta)=-\int_0^\theta\log|2\sin u|\,du. \] No factor in this range vanishes. Uniformly in $k$, \begin{equation}\label{wsref-0-61-34-33-31-3a-66-61-63-74-6f-72-69-61-6c} \frac{2\pi}{r}\log F_r(k) =-\Lambda(2\pi k/r)+O(\log r/r). \end{equation} To prove this, apply a Riemann-sum comparison to $f(t)=\log(2\sin 2\pi t)$ on $(0,1/2)$. The function increases up to $1/4$ and decreases afterwards. On either monotone portion the difference between the right-endpoint sum and $r$ times the integral is bounded by endpoint oscillations. The first interval adjoining zero is handled separately using $f(t)=\log t+O(1)$ there; its error is bounded. At the other end the last possible sample is at distance $1/(2r)$ from $1/2$, so its absolute value is $O(\log r)$. Splitting at the maximum, with at most one crossing cell, gives \[ \sum_{j=1}^kf(j/r)=r\int_0^{k/r}f(t)\,dt+O(\log r). \] The change of variable $u=2\pi t$ proves (a431:factorial). The logarithmic endpoint singularities have been controlled rather than discarded by an unjustified uniform-continuity claim.

In the convention $[j]_q=(q^j-q^{-j})/(q-q^{-1})$ at this root, \[ [k]_q!= \frac{F_r(k)}{(2\sin(2\pi/r))^k}. \] For a product of factorials and inverse factorials whose signed indices sum to zero, the denominator powers cancel exactly. Formula (a431:factorial) then gives a finite sum of Lobachevsky terms with error $O(\log r/r)$, provided the number of factors is fixed and all indices lie in the displayed range. Other index ranges or quantum-symbol conventions require their own sign and zero-factor checks. This building-block calculation is not yet an asymptotic formula for a complete $6j$-symbol.

A Laplace principle with all hypotheses visible.

Consider finite label sets $K_r$ in a compact set $K$, with $|K_r|\leq r^C$, positive weights $W_r(x)$, and a continuous action $S:K\to\mathbb R$. Suppose \[ \sup_{x\in K_r} \left|\frac{2\pi}{r}\log W_r(x)-S(x)\right| \longrightarrow0. \] Suppose also that for each sufficiently large odd $r$ there is $x_r\in K_r$ tending to a maximizer of $S$. Then \begin{equation}\label{wsref-0-61-34-33-31-3a-6c-61-70-6c-61-63-65} \lim_{\substack{r\to\infty\\r\ {\rm odd}}} \frac{2\pi}{r}\log\sum_{x\in K_r}W_r(x) =\max_KS. \end{equation} For the upper bound use the largest term times $|K_r|$; its extra normalized logarithm is at most $2\pi C\log r/r$. For the lower bound retain the single term $W_r(x_r)$. These two estimates prove the claim and show exactly where positivity is used.

A statement only along some subsequence of odd levels would not give the target limit. Admissible labels must approach a maximizer at every sufficiently large odd level in the argument just given. Uniformity near boundary labels also matters: pointwise asymptotics at each fixed shape do not control a maximum that moves toward a singular boundary as $r$ grows.

The geometric maximization in one ideal tetrahedron.

For an ideal hyperbolic tetrahedron with positive dihedral angles $\alpha,\beta,\gamma$ summing to $\pi$, the established volume formula [E2] is \[ V(\alpha,\beta)= \Lambda(\alpha)+\Lambda(\beta)+\Lambda(\pi-\alpha-\beta). \] Its derivatives are explicit: $\partial_\alpha V=\log(\sin\gamma/\sin\alpha)$, and similarly for $\beta$. Its negative Hessian is \[ \begin{pmatrix} \cot\alpha+\cot\gamma&\cot\gamma\\ \cot\gamma&\cot\beta+\cot\gamma \end{pmatrix}. \] The first diagonal entry is $\sin\beta/(\sin\alpha\sin\gamma)>0$, and its determinant equals one, using $\alpha+\beta+\gamma=\pi$. Thus $V$ is strictly concave on the angle triangle. The only interior critical point is $\alpha=\beta=\gamma=\pi/3$. The volume extends to zero at the boundary, so this is its unique maximum, of value $v_3=3\Lambda(\pi/3)$.

This is an exact geometric check of a potential action, not a claim that every quantum $6j$-symbol has this particular domain or this upper bound. Quantum-symbol asymptotics can involve different polyhedra and different angle regimes. A proof for a manifold must identify the actual action and impose its gluing constraints; adding independently maximized tetrahedral volumes would generally ignore those constraints.

Cancellation is the first missing lower-bound mechanism.

A state sum may have signed or complex summands even when the total invariant is nonnegative in an applicable normalization. For general summands $z_{r,x}$ the triangle inequality gives an upper bound from $\max|z_{r,x}|$, but no lower bound. The two terms \[ e^{rA},\qquad -e^{rA}+1 \] both have exponential size $e^{rA}$ for $A>0$ while their sum is one. Positivity of the final sum does not fix this cancellation. Squaring an amplitude also does not undo cancellation already present inside it.

A sufficient repair would be an estimate \[ \left|\sum_xz_{r,x}\right| \geq e^{-o(r)}\max_x|z_{r,x}|, \] together with the correct action asymptotic. Alternatively one could find a genuinely positive expansion to which (a431:laplace) applies. Neither is established here for arbitrary manifolds. This is why knowing the largest factorial ratio or a stationary critical shape does not by itself prove the conjecture.

For a proposed positive expansion there are still geometric obligations: identify its maximal action with the complete hyperbolic volume, verify admissibility at every large odd level, and rule out other label regimes with larger action. A formal critical-point equation alone proves none of these global comparisons.

Verification, gap, and outcome.

The factorial normalization and its balanced cancellation were checked at finite odd levels, including indices near both logarithmic endpoints. The tetrahedral Hessian identity is exact; numerical values only supplement it. No computed Turaev--Viro values or finite range of levels is used as proof of an asymptotic limit.

The first missing input is a uniform asymptotic and noncancellation mechanism for the actual invariant in the stated manifold classes, followed by the correct global geometric maximization. UNRESOLVED. The building-block asymptotic, conditional Laplace principle and ideal-tetrahedron maximization are proved. They do not establish the Chen--Yang limit in general or justify a change of root or normalization.

Sources

• S1 Qingtao Chen and Tian Yang, Volume conjectures for the Reshetikhin-Turaev and the Turaev-Viro invariants, v4 $2018$. https://arxiv.org/html/1503.02547. Formulation source.
• E1 Renaud Detcherry, Efstratia Kalfagianni and Tian Yang, Turaev--Viro invariants, colored Jones polynomials and volume. https://arxiv.org/abs/1701.07818. Primary source for specified link-complement results, not unrestricted volume convergence; consulted 27 September 2026.
• E2 John W. Milnor, Computation of volume, Chapter 7 of William Thurston's The Geometry and Topology of Three-Manifolds, 1980 notes, electronic edition 1.1 $2002$, especially Theorem 7.2.1. https://library.slmath.org/nonmsri/gt3m/PDF/7.pdf. Primary text for the ideal-tetrahedron volume formula; consulted 27 September 2026. The Hessian and maximum are computed explicitly above.