MathDB

Chen–Yang volume conjecture

For every closed orientable hyperbolic 33-manifold MM, the odd-level SO(3)SO(3) Turaev–Viro invariants satisfy

lim⁡r→∞r odd2πrlog⁡∣TV⁡rSO(3)(M)∣=Vol⁡(M).\lim_{\substack{r\to\infty\\ r\ \mathrm{odd}}}\frac{2\pi}{r}\log\left|\operatorname{TV}^{SO(3)}_r(M)\right|=\operatorname{Vol}(M).
References

Primary source

arXiv

Additional references

3-manifold-topologyquantum-topology
Posted · edited

Progress summary

Refreshed
Claimed solved

An unrefereed preprint claims the conjecture for a broad class of sufficiently long integral fillings, but this has not been independently verified.

The Chen–Yang volume conjecture relates the growth of quantum invariants to hyperbolic volume for Dehn fillings. The latest claim concerns fundamental shadow links and sufficiently long integral fillings.

Known results

  • Numerical tests in 2021 supported the predicted growth for selected knot exteriors in handlebodies, with possible numerical errors in some cases.
  • A 2022 cabling theorem showed that the asymptotic relation is preserved under specified cable operations.
  • A 2022 paper described a first-principles perturbative explanation, but the supplied record does not establish its scope or proof status.

September 10, 2026 long-fillings claim

Ce Shen's preprint claims that exponential growth of SO(3)\mathrm{SO}(3) Turaev–Viro invariants equals hyperbolic volume for sufficiently long integral fillings of fundamental shadow links. This is substantial progress toward the conjecture, but the result is unrefereed and does not settle every case.

Current status (as of September 2026): A claimed unrefereed result covers sufficiently long integral fillings of fundamental shadow links; the full conjecture remains unverified and open beyond that regime.

Sources

Solutions 0

No solutions have been posted yet.