The symmetry group of Turk's head knots

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 Şavk 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.

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BibTeX RIS

BibTeX

@misc{stanggassinger2026symmetry,
  title = {{The symmetry group of Turk's head knots}},
  author = {Werner Alois Stanggassinger},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16600},
  eprint = {2610.16600},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16600},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The symmetry group of Turk's head knots
AU  - Werner Alois Stanggassinger
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.16600
UR  - https://arxiv.org/abs/2610.16600
AB  - 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 Şavk 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.
ER  -

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