Torus-knot sculpture

A tube follows a (p, q) torus knot — a curve that winds p times around an imaginary torus of radius R while threading q times through its hole:

C(u) = ((R + a cos qu) cos pu, (R + a cos qu) sin pu, a sin qu), u ∈ [0, 2π)

The tube is swept with a numeric Frenet frame: T = ∂C/∂u, N from the change of T, and B = T × N, so P(u,v) = C(u) + rt(cos v N + sin v B). The twist slider rotates that frame by twist · u as it travels, which keeps the surface closed and adds flow.

When gcd(p, q) = 1 the curve closes into a single knot; otherwise it forms a link of gcd(p, q) separate components. The tube radius is held below a/2 so the crossings keep generous separation, and a single material keeps the geometry the subject.