The fibered-knot coefficient conjecture for theta

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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)∣T→T1.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.

References

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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