Positive amphichirality of Turk's head knots: the orders of the maps

Let p >= 3 be odd, q >= 3 and gcd(p,q) = 1. We show that the Turk's head knot Th(p,q) is strongly positive amphichiral if and only if q is odd, and that the orders of the finite-order diffeomorphisms realising its positive amphichirality are exactly the numbers 2q/d, d an odd divisor of q. This answers Problem D of Di Prisa and Savk (arXiv:2409.20106) for these knots. The proof reads the answer off the symmetry group D_2q determined in a companion paper (arXiv:2610.16600), together with the action of each symmetry on the orientation of the knot. For q = 2 and p >= 5 odd we show that the two positive amphichiral classes in the symmetry group D_4 of Th(p,2) have order 4; hence every finite-order homeomorphism realising the positive amphichirality of Th(p,2) has order divisible by 4, and Th(p,2) is not strongly positive amphichiral even with topological involutions. Moreover, every finite-order diffeomorphism realising it has order exactly 4; we do not show that such a diffeomorphism exists. For p = 3, the figure-eight knot, only the piecewise linear non-strong-positivity of Hartley and Kawauchi is available here, and the orders remain conditional. The q = 2 part uses our note doi:10.5281/zenodo.23562743. The deposit contains the note (PDF and LaTeX source) and the scripts of its checks with their outputs; the proofs do not rest on computation. STATUS: not refereed. 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. Details are in the section on tool use at the end of the note. Licences: text CC BY 4.0; code GNU General Public License v3.0 or later; data CC BY 4.0.