Paper · Preprint · deposited 2026-10-22

The symmetry group of the Turk's head knots Th(p,2)

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

Full text (PDF)

Permanent archive (all files, all versions): doi:10.5281/zenodo.23562743 (Zenodo, CERN).

Abstract

Di Prisa and Savk conjectured that the Turk's head knot Th(p,q), gcd(p,q) = 1, has symmetry group the dihedral group of order 4q (Conjecture R of their survey, arXiv:2409.20106). The case q >= 3 is proved in arXiv:2610.16600, which excludes q = 2 because the standard diagram has bigons. This note treats q = 2: for every odd p >= 3, Th(p,2) is a hyperbolic, amphicheiral two-bridge knot, and hence its symmetry group is the dihedral group of order 8, as conjectured. The proof combines three known facts with one observation, that Th(p,2) is a two-bridge knot; for p = 3 (the figure-eight knot) the theorem was already known. The proof does not use any computer result. The deposit also contains four scripts with their outputs, independent numerical checks described in Section 4 ('Checks') of the note: for odd p <= 41 both checkerboard (Tait) graphs of the closed braid diagram are isomorphic to the graph predicted in Section 2; the rotation condition used for the Conway circles (odd 5 <= p <= 41); for odd p <= 15 and q = 2, 3 the word identity used for amphicheirality, signature 0 and V(t) = V(1/t); and for odd p <= 21 that the determinant is a Pell number, that SnapPy finds an isometry with the two-bridge knot C(2,...,2) and the symmetry group D4. The SnapPy isometries and symmetry groups are numerical and not interval-verified. CONTENTS: the note (PDF and LaTeX source), the scripts, their outputs and NAME_MAP.md (every file with its working path and a checksum). HONESTY NOTES: no invariant is computed by an AI; the checks were run in Python with SnapPy 3.3.2 and spherogram inside SageMath 10.9 and with NetworkX. This work was developed in collaboration with the AI system Claude Code (Anthropic): the choice of problem, the direction of the work and the computing hardware are due to the author (a sculptor); the formalisation, the algorithms and the proofs were developed by Claude Code (Anthropic) working with the author. The collaboration is stated as a fact, not a disclaimer; details are in the section on tool use at the end of the note.

Keywords: knot theory · Turk's head knots · two-bridge knots · amphicheiral knots · hyperbolic knots · symmetry group · dihedral group · Conjecture R

Cite

@misc{stanggassinger2026symmetry,
  author    = {Stanggassinger, Werner Alois},
  title     = {The symmetry group of the Turk's head knots Th(p,2)},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23562743},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/symmetry-group-turks-head-knots-th-p-2/}
}

This page mirrors the Zenodo record; if the two ever differ, the Zenodo record is authoritative.