Paper · Preprint · version 3 of 2026-10-22, first deposited 2026-10-06

Conway polynomials of Th(p,q) and Th(q,p): a proof of Takemura's conjecture

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

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Permanent archive (all files, all versions): doi:10.5281/zenodo.23183920 (Zenodo, CERN).

Also on arXiv: arXiv:2610.32269 [math.GT], v1 of 2026-10-23.

Abstract

Takemura conjectured that the Conway polynomials sum a_l z^l of the Turk's head link Th(p,q) and sum b_l z^l of Th(q,p) satisfy a_l = b_l for l = 0,1 (mod 4) and a_l = -b_l for l = 2,3 (mod 4), and proved it for q = 2, 3. We prove it for all p, q >= 1, knots and links. Equivalently, nabla_Th(q,p)(z) = c nabla_Th(p,q)(iz) with c = 1 if gcd(p,q) is odd and c = -i if it is even. The proof combines a product formula nabla_Th(p,q)(z)^2 = prod_{j=1}^{p-1} prod_{k=1}^{q-1} (z^2 + 2cos(2 pi k/q) - 2cos(2 pi j/p)), which comes from the reduced Burau representation, with the leading coefficients, read off the Seifert surface of the closed braid. The deposit contains the note (PDF and LaTeX source) and the scripts and outputs behind its Section "Checks".

Keywords: knot theory · Conway polynomial · Alexander polynomial · Turk's head knots · Takemura's conjecture · reduced Burau representation · weaving knots

Cite

@misc{stanggassinger2026takemura,
  author    = {Stanggassinger, Werner Alois},
  title     = {Conway polynomials of Th(p,q) and Th(q,p): a proof of Takemura's conjecture},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23183920},
  eprint    = {2610.32269},
  archivePrefix = {arXiv},
  primaryClass = {math.GT},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/takemura-conjecture-turks-head-knots/}
}

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