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Sierpinski Knots

We can build families of wild knots by joining tame knots recursively. Here is one nice example I call the Sierpinski Trefoil.

Start with a trefoil--

Form a cyclic twisted sum of three trefoils--

Form a cyclic twisted sum of three of these--

And just keep going and going and going....

We can associate a symbol sequence with the "edges" of the knot as we traverse the knot by following these rules:

Number the triangles of each level of the Sierpinski knot construction counterclockwise from the top as 1 2 3

Choose a starting point and follow the knot, assigning 1, 2 or 3 according to which side you have passed, with one symbol for each level.
For example, the first knot has the symbol sequence, starting at the apex and moving counterclockwise, 11,23,32,33,12,21,22,31,13
and the second has the sequence 111,123,132,313,331,322,233,212,221,222,231,213,121,112,133,311,323,332,333, 312,321,232,223,211,122,131,113. At each level n of the construction an 'edge' of the knot is coded by a sequence of length n+1. In the limit each 'edge' of the knot corresponds to an infinite sequence on three symbols, and the knot induces an ordering on the symbol sequences.

Question--what is the Conway, Jones or bracket polynomial of this knot in terms of n?

Below is the knot in the fourth stage of construction showing three paths that snake through the knot until right before they self-intersect. Note the similarity to the beginnings of a Hilbert or Sierpinski space filling curve. In this case, the superposition of three 'space filling' curves forms a universe of the knot diagram with Hausdorff dimension log4/log3 (in the limit), and the weaving together of the three curves forms a fractal knot.

But wait! There's more! Notice that any two of the curves are not woven, that is, if you removed one of the curves from the knot, the other two would fall apart, like Borromean Rings (or in this case, Borromean segments).

If we do not twist the strands before joining the tame knots, we have, in the case of the trefoil, a wild link. We can build wild knots and links like this from any knotted knot, not just the trefoil.

Alternatively, we can weave this knot from the unknot if we are allowed to pass strands through one another. The figure below shows the generating rule for the deformation of an "edge" of the knot at some stage of the construction.

We only have to pass one crossing per triangle to pass to the next level of the knot if we weave it. This is sort of like a three handed infinite fingered fourth dimensional cat's cradle construction. The strand switching in this example takes place at the bottom crossing.