Frenet turns

We discuss a problem posed by A. Agrachev asking howmany times a usual circle in Rn should be traversed to admit a defor-mation by curves with nowhere degenerating Frenet frame. It turns outthat the answer depends on a specific topology which we consider. Forthe literal Cn curve topology, the least number of turns of a plane circleadmitting arbitrarily small nondegenerate perturbations isk(2) = 1, k(3) = 2, k(n) = 1 (n ≥4).This jet-level problem is different from the original Frenet-controlproblem by Agrachev. We show that in the literal interpretation ofAgrachev’s problem one has a simple spherical Fenchel obstruction inall dimensions n ≥4.To retain a nontrivial turn-counting problem, we introduce decoratedturn data. In R4 the datum is a pair (p, q) recording tangent-plane andnormal-plane turns; we prove that every nonresonant pair (p, q) withp, q > 0, p ̸= q, is accessible by small positive constant Frenet con-trols. In even dimension 2r the analogous datum is a vector (p1, . . . , pr ),and every vector with pairwise distinct positive entries is accessible byconstant controls. Odd dimensions require genuinely time-d