Mathematics · Study 061
Torus Knot
A rotating 3D torus knot with proper over and under crossings. Click through the trefoil, the cinquefoil and six more knots.
What you're looking at
A torus knot winds p times around the axis of a doughnut while passing q times through its hole. The numbers must share no common factor or the rope splits into separate loops. The curve is computed in three dimensions, and nearer strands are drawn thicker with a gap cut beneath them so the crossings read clearly.
The idea behind it
The simplest case, 2 and 3, is the trefoil, the most basic knot that cannot be untied. Swapping p and q gives the same knot laid differently on the torus. Knot theory began in the 1800s with a failed theory of atoms as knotted vortices, and now informs the study of DNA and polymers.
Try this
Move the pointer to turn the knot. Click or tap for the next pair of winding numbers.
