Paper · Preprint · version 4 of 2026-10-22, first deposited 2026-10-05

Hoste's condition for all Turk's head knots

Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)

Full text (PDF)

Permanent archive (all files, all versions): doi:10.5281/zenodo.23154788 (Zenodo, CERN).

Abstract

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".

Keywords: knot theory · Alexander polynomial · Turk's head knots · Hoste's conjecture · reduced Burau representation · weaving knots · zeros of polynomials

Cite

@misc{stanggassinger2026hoste,
  author    = {Stanggassinger, Werner Alois},
  title     = {Hoste's condition for all Turk's head knots},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23154788},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/hoste-condition-turks-head-knots/}
}

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