The ADO radial-limit formula for nice knots

About 5 years old · traced to

Let KK be a nice knot, let FK(x,q)F_K(x,q) be its FKF_K series, let Np(K,x)N_p(K,x) be its ADO polynomial, let ΔK(x)\Delta_K(x) be its Alexander polynomial, and let ζp=e2πi/p\zeta_p=e^{2\pi i/p} be a root of unity. ADO radial-limit conjecture.

FK(x,q)x1/2−x−1/2∣q=ζp=Np(K,x)ΔK(xp).\left.\frac{F_K(x,q)}{x^{1/2}-x^{-1/2}}\right|_{q=\zeta_p}=\frac{N_p(K,x)}{\Delta_K(x^p)}.

This is the precise nice-knot version of the radial-limit relationship attributed to Gukov and Nakajima. The source notes that the relationship may fail for knots that are not nice, so the general case remains open.

References

Primary source

Paul Orland, Lara San Martín Suárez, Toby Saunders-A'Court and Josef Svoboda, “Quantum Invariants and Fiberedness”, arXiv:2512.23700 (2026).

Additional references

2 papers in this index state this conjecture (2021–2025). The statement above is taken from the most recent of them; the others are arXiv:2107.14238.

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