Torus knots

Drag p and q: the line winds on a torus, p times round its axis and q times round its tube. When p and q have no common factor but 1 it is a single closed line, a knot: with p or q equal to 1 it is not knotted at all, and p = 2, q = 3 is the trefoil. The corner counts the crossings in the view straight down the torus's axis, q(p − 1) of them (the picture itself is oblique and turning, so you may not see that many); Kunio Murasugi proved in 1991 that however the knot is drawn it has at least min(p(q − 1), q(p − 1)) crossings. When p is no larger than q, the view from above already shows the fewest; when p is larger, the extra crossings go away in another drawing: T(p, q) and T(q, p) are the same knot. When p and q share a factor greater than 1 the line never closes up as one: with highest common factor d it is d separate loops, each its own colour, each a T(p/d, q/d), and every two are linked pq/d² times, measured in the corner with Gauss's integral along the loops.