The fibered-knot coefficient conjecture for theta

Let KK be a fibered knot, and let dd be the degree of its Alexander polynomial Δ(K)\Delta(K), meaning the highest power of TT. Consider the coefficient of T22dT_2^{2d} in θ(K)\theta(K), which is a polynomial in T1T_1.

Fibered-knot coefficient conjecture. This coefficient is an integer multiple of

T1dΔ(K)TT1.T_1^d\left.\Delta(K)\right|_{T\to T_1}.

The claim would provide a fast-to-compute fiberedness criterion and can be stronger than the Alexander condition. The authors state that they have no theoretical support for it, so it remains open.

Sources & referencesView supporting material

Primary source

Dror Bar-Natan and Roland van der Veen, “A Fast, Strong, Topologically Meaningful and Fun Knot Invariant”, arXiv:2509.18456 (2026).

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