Drinfeld Twists Between Non-Isomorphic Rack Solutions: A Negative Answer to a Question of Ferri

In an extended abstract in Oberwolfach Report 49/2025 (Oberwolfach Rep. 22 (2025), no. 4, 2633–2672; pp. 2645–2649), D. Ferri proposes to classify set-theoretic solutions of the Yang–Baxter equation up to Drinfeld twists, and asks whether two rack solutions on the same set that are related by a Drinfeld twist must coincide. Isomorphic solutions are always related by a twist (Ghobadi, Ferri), so the question is whether such rack solutions must be isomorphic. We show that they need not be. On X = Z/5 the rack solutions r(a,b) = (2b − a, a) and s(a,b) = (3a + 3b, a) of the affine quandles Aff(Z/5,2) and Aff(Z/5,3) are related by an explicit Drinfeld twist, while the two quandles are not isomorphic: the second is commutative and the first is not. Every step is an explicit finite identity, and an exact finite computation confirms all of them. More generally, Aff(Z/p,t) and its dual Aff(Z/p,t⁻¹) are non-isomorphic racks with twist-related rack solutions whenever −t has multiplicative order 3, 4 or 5 modulo the prime p; this gives such pairs of infinitely many orders. The proofs use Ferri's criterion in terms of the braid group B₃, Burnside's theorem on marks and the finite von Dyck groups. By exact computations, 5 is the smallest order of such a pair, there are 16 such pairs of order 6, and Aff(Z/7,2) is not twist-related to its dual. Status: the definite question answered negatively; the classification programme open. The classification of solutions up to Drinfeld twists stays open, and we do not address it. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-14299580-006 (Ferri, Drinfeld twists for solutions to the Yang–Baxter equation, Oberwolfach Rep. 22 (2025), no. 4, 2633–2672, pp. 2645–2649).