The fibered-knot leading-term conjecture for FKF_K

Let KK be a fibered knot, let \genusK\genusK be its genus, let \hopfK\hopfK be its Hopf invariant, and let FKF_K have leading term determined by the exponent \el(K)\el(K). Fibered-knot leading-term conjecture. The knot KK is nice and

\el(K)=\genusK\hopfK,\el(K)=\genusK-\hopfK,

so that

lt(FK)=q\genusK\hopfKx\genusK1/2.\operatorname{lt}(F_K)=q^{\genusK-\hopfK}x^{\genusK-1/2}.

This packages the conjectured characterization of fibered knots with the predicted leading behavior of their FKF_K invariant. It is proved only in the classes covered by the preceding theorems and is open in general.

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).

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