Torus Knots as Braid Closures: Crossing Number & Writhe

This is the 2D companion to the 3D torus-knot simulator, which projects a rotating 3D curve to show that writhe depends on viewing angle. Here the same knot type is built from an entirely independent 2D construction: the standard braid word (σ₁σ₂⋯σ_(p−1))^q on p strands, closed into a diagram by joining each bottom strand back to its matching top strand. Every crossing in the rendered diagram, its sign, and the resulting writhe are counted directly from that generated word — no curve is projected and no camera exists. Because the diagram's crossing count is exactly q(p−1), it is only the true topological minimum (Murasugi's min(p(q−1), q(p−1)) formula) when p ≤ q; swapping p and q redraws the identical knot T(p,q) = T(q,p) with a different crossing count, making concrete — through a completely different mechanism than camera rotation — that crossing number and writhe belong to the diagram, not to the knot.