Torus knots

Cylinder Torus   P=12 + –   Q=12 + –

The surface shown can be a torus (a doughnut-shaped surface) or a cylinder, depending on the position of the cursor above.

A certain number of segments appear on the cylinder (which become a – possibly different – number of closed curves as the cylinder closes into a torus).

The number and shape of the curves depend on the parameters P and Q. If Q is 0, the curves in the figure are P circles on the torus (and P segments on the cylinder). If Q is different from 0, one or more knots are formed (one for each of the colors involved). Can you predict how many (depending on P and Q)?

Click & drag on the image to change the point of view.

On the cylinder, the P parameter controls the number of curves that intersect a given “meridian”, while the Q parameter controls the slope of the curves.

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