Published October 22, 2026 | Version v4
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Hoste's condition for all Turk's head knots

Description

We show that every zero z of the Alexander polynomial of every Turk's head knot Th(p,q) satisfies Re z > -1, and that every non-real zero lies on the unit circle. This answers Problem H of Di Prisa and Savk, which asks to verify Hoste's condition for all Turk's head knots; the survey records it for p = 3. The proof is an exact factorisation of det(I - x S(sigma_1 sigma_2^-1 sigma_3 ... sigma_{p-1}^{+-1})) for the reduced Burau representation S into floor((p-1)/2) quadratic factors in t, whose middle coefficients involve c_j = 4cos^2(pi j/p), and possibly one linear factor, followed by an elementary root estimate. The deposit contains the note (PDF and LaTeX source) and the scripts and outputs behind its Section "Checks".

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Additional details

Is supplemented by
Other: https://knot-structures.stainlesssteel4u.de/ (URL)
References
Preprint: arXiv:2409.20106 (arXiv)
Preprint: arXiv:1506.02000 (arXiv)
Preprint: arXiv:2606.15256 (arXiv)
Preprint: arXiv:2606.11301 (arXiv)
Preprint: arXiv:2607.14638 (arXiv)
Preprint: arXiv:2506.17889 (arXiv)

References

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  • A. Di Prisa and O. Şavk, Turk's head knots and links: a survey, arXiv:2409.20106v3 (2026). Journal version: Expo. Math. 44 (2026), 125777.
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