Twist and Writhe of Curves on a Torus: An Exact Formula, Two Families, and the Clifford-Torus Symmetry

Curves that wind on a torus occur in DNA toroidal condensates, in the magnetic surfaces of toroidal plasmas and in knotted light, and the way their linking number divides into twist and writhe is usually obtained numerically. For (p,q) curves of the linear family, in which the poloidal angle advances uniformly with the toroidal angle, we give the twist relative to the surface normal in closed form as a complete elliptic integral of the first kind; the writhe follows from Wr = pq − Tw. For the conformal family, the stereographic images of uniform curves on tori in S³ (loxodromes of the torus), the twist reduces to the elementary expression Tw = pq/√(p²(1−ε²)+q²ε²), where ε = r/R. Two consequences follow. (i) In the conformal family the exchange (p,q) ↔ (q,p) leaves the writhe unchanged provided the aspect ratio is exchanged as ε ↦ √(1−ε²); this holds for every ε. (ii) In the linear family the exchange symmetry Wr(p,q) = Wr(q,p) holds for every winding ratio, rational or not, if and only if R/r = √2, the Clifford torus; we give a short two-sided proof. We show that the field lines of any axisymmetric magnetic field whose flux surface is a circular-section torus belong to the conformal family, and that a single Shafranov-type gradient δ connects the two families (δ = 0 conformal, δ = −ε linear). The exchange defect grows as δ² between paired tori and as δ³ on the Clifford torus; we prove both orders by showing that the conformal family is a stationary point of the twist along the bridge, and give the leading cubic coefficient on the Clifford torus in closed form, c₃ = (15/4)(3√2 − 4) p³q³(q² − p²)/(p² + q²)^(7/2). The defect is a smooth function of the winding ratio; to leading order its maximum lies at λ* = √((9 + √57)/6) ≈ 1.661, a root of 3λ⁴ − 9λ² + 2 = 0, and not at the golden ratio. Applied to the twisted-toroid model of DNA condensates, the formula predicts an internal layer in which the twist per meridional turn equals one; such a layer exists if and only if 0.595 < k < 1, and it coincides with the Clifford layer at k = 0.745. All results are checked by independent quadratures and by the integer condition |Tw + Wr| = pq.