Torus Knots: Writhe & Crossing-Sign Explorer

This simulator renders a real torus knot T(p, q) as a 3D tube and computes its diagram invariants live, straight from the geometry, instead of looking them up in a table. Every frame the sampled curve is projected along the current camera direction, every pair of strands is tested for an actual 2D crossing, and each crossing is signed by comparing camera-space depth (which strand is "over") against the handedness of the two tangent directions — exactly the construction behind the Gauss linking integral and the writhe of a knot diagram. Rotating the same, unmoving curve changes the crossing count and the writhe, making concrete the fact that writhe is a property of a diagram's projection, not of the knot itself — unlike the crossing number's coprime-torus-knot formula min(p(q−1), q(p−1)) or the Alexander polynomial, which stay fixed under any Reidemeister-equivalent redrawing.