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 \(X=\mathbb{Z}/5\) the rack solutions \(r(a,b)=(2b-a,a)\) and \(s(a,b)=(3a+3b,a)\) of the affine quandles \(\operatorname{Aff}(\mathbb{Z}\)\(/5,2)\) and \(\operatorname{Aff}(\mathbb{Z}\)\(/5,3)\) 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, \(\operatorname{Aff}(\mathbb{Z}\)\(/p,t)\) and its dual \(\operatorname{Aff}(\mathbb{Z}\)\(/p,t^{-1})\) 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 \(B_3\), 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 \(\operatorname{Aff}(\mathbb{Z}\)\(/7,2)\) 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.