The ADO radial-limit formula for nice knots

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)x1/2x1/2q=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.

Sources & referencesView supporting material

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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