Torus Knot

Back to interactive Torus Knots

These knots are called torus knots because they can be realized on a torus. No knot can be realized on a sphere.

All yellowish pictures are anaglyphs. They are computed with small eye separation so that one can also view them, non-stereo, without red-green glasses.

torus knot 19 2 ana
This 19-2-torus knot can easily be imagined to lie on a torus.
torus knot 5 2 ana
Note that this and the previous knot are “alternating”, which means: underpasses and overpasses alternate. An alternating knot cannot be realized with fewer crossings if the knot is without an “isthmus”. A crossing is called an 'isthmus', if a (diagonal) cutting through the crossing decomposes the knot into two pieces.
torus knot 7 3
This 7-3-knot is not alternating since the underpasses and overpasses occur in pairs.
knot2 3to3 2 001
This animation shows how a 3-2-knot is deformed into a 2-3-knot. We use the 'conformal inside-out rotation' which is explained for the Clifford Torus .
torus knot elast 001
These knots are computed as if the square rod were elastic and the color bands were straight before the rods were bent into these knots.
torus knot oscc 3 2
This 3-2-knot is shown with 'osculating circles', these circles have the same curvature as the space curve has at the points of contact.
torus knot rot7 2 001
A knot which looks rather simple in its most symmetric form may look much more complicated when viewed from a different direction.