This simplest of all knots, known as the "trefoil", readily lends itself to iteration due to its simplicity and three-fold symmetry.
One possible starting self-similar knot:
After one iteration:
After three iterations:
Flash animation illustrating that the second iteration of this version of an iterated trefoil knot is unicursal.
An alternative approach for this knot is to first rearrange it into a figure-8 configuration. The iterative procedure now doubles the number of copies with each generation, as opposed to the tripling in the triangular version above.
The starting self-similar knot:
After the first iteration:
Flash animation illustrating that the third iteration of this version of an iterated trefoil knot is unicursal.
To see a digital artwork based on this iterated knot, click on the thumbnail below.
A 3-dimensional iterated trefoil knot is shown below. These figures were created using KnotPlot, which allows manipulation of the knots in 3-D. The KnotPlot files are available by clicking on the links below.
The starting knot:
The KnotPlot file for the starting knot
The knot after one iteration:
The KnotPlot file for the knot after one iteration
The starting knot:
The KnotPlot file for the knot after two iterations