Paper · Preprint · version 2 of 2026-10-26, first deposited 2026-10-08

The symmetry group of 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.23234787 (Zenodo, CERN).

Also on arXiv: arXiv:2610.16600 [math.GT], v2 of 2026-10-12.

Abstract

For p, q >= 3 with gcd(p,q) = 1 we show that the symmetry group of the Turk's head knot Th(p,q), the closure of the braid (sigma_1 sigma_2^{-1} sigma_3 sigma_4^{-1} ... sigma_{p-1}^{+-1})^q, is the dihedral group of order 4q. This proves Conjecture R of Di Prisa and Savk for q >= 3; the case p = 3 is due to Sakuma and Weeks. The argument is diagrammatic: by the flyping theorem of Menasco and Thistlethwaite, the symmetries of a knot with a prime, reduced alternating diagram without non-trivial flypes are the symmetries of the diagram (we write out a proof for the diagrams at hand), and we show that the standard diagram of Th(p,q) has no non-trivial flypes and exactly 4q symmetries. The chirality of Th(p,q) for even p, which also follows from a theorem of Murasugi, is visible in the group. We compare the argument with the geometric one of Sakuma and Weeks for p = 3. This is the version posted on arXiv as arXiv:2610.16600. The deposit contains the paper (PDF and LaTeX source, the source identical to arXiv:2610.16600v2) and the scripts and outputs behind its Section 8 ("Computational checks").

Keywords: knot theory · Turk's head knots · weaving knots · symmetry group · alternating knots · flyping theorem · Tait graph · dihedral group

Cite

@misc{stanggassinger2026symmetry,
  author    = {Stanggassinger, Werner Alois},
  title     = {The symmetry group of Turk's head knots},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23234787},
  eprint    = {2610.16600},
  archivePrefix = {arXiv},
  primaryClass = {math.GT},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/symmetry-group-turks-head-knots/}
}

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