Turk's head knots with at least three leads and three bights are not perfectly straight

Owad defined the straight number str(K) of a knot, the least number of crossings of a diagram in which one straight strand meets all crossings, and called K perfectly straight if str(K) equals the crossing number. Di Prisa and Savk asked whether the Turk's head knot Th(p,q) is perfectly straight when p >= 3 and q < p (Problem A of their survey, arXiv:2409.20106). We prove that Th(p,q) is not perfectly straight for all p, q >= 3 with gcd(p,q) = 1. For q < p this answers Problem A when q >= 3; for q > p it is a theorem of Owad, of which we give an independent proof. The proof combines an exact count on the standard diagram (its longest repetition-free run of crossings has length 2q+p-2*floor(q/p)-3, less than the crossing number (p-1)q) with the uniqueness of the shadow of the minimal diagram up to homeomorphism of the sphere, which follows from the flyping theorem. For q = 2 we show, using the two-bridge lemma of a separate note of ours (doi:10.5281/zenodo.23562743) and a theorem of Belousov and Malyutin, that Th(p,2) is perfectly straight for every odd p >= 3. Using that lemma, this answers Problem A for 2 <= q < p: for p >= 3 and 2 <= q < p, Th(p,q) is perfectly straight if and only if q = 2. The proofs use no computer result; the deposit also contains the two scripts of the section 'Checks' and their outputs. 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.