In my last post I presented the concept of the twisted knot, which itself is specific idea to describe a molecular knot, or what a molecular bond really is (at least according to my current hypothesis).
In it I showed a nice tight twisted knot:
However, the problem with such a knot is that it doesn’t seem a good explanation why the structure of the knot looks so different to the strings outside of the knot. At first, I considered the possibility that the knot indeed looks different than the rest of the orbital (i.e. the curved string). However, the more I thought about it, the more likely it seems that the geometry of the strings of dots in the knot are more or less identical to what it is elsewhere.
To test this idea, I made a helix with a radius to height ratio of 1 to 100 and bent it 45 degrees. Then I made duplicates so that there were four of them and rotated them by 90, 180 and 270 degrees. And this is the shape that I got:
The thickness of the string was made to be equal to the radius of the unbent helix. With this amount of bending, when the helices were placed as close to each other as possible, a bulge appeared at the center. However, if we ignore the bulge, the rough shape of the twisted strings seems somewhat reasonable.
Next, I attempted to replicate the shape with two saint Hannes knots. While I can’t say that the outcome was identical, it isn’t too far off either:
This seems to indicate that the main idea is sound. There are still quite a few problems to be solved. The biggest of them is the bulge that really shouldn’t be there.
So, at least for now, let’s continue with the twisted knot, but instead of trying to attach circular arcs at either end, we can just add identical helices as in the central knot, but just bending the helices just enough for them to no longer knot:
Granted, the above picture still looks a bit off. But you can clearly see that there are strings flaring to eight symmetrical directions from the twisted knot, just as the combination of two saint Hannes knots requires.
And as I look at the pictures above, it seems that the twisted knot is a sort of ‘unnatural shape’ to add to the saint Hannes knot. That is, the twisted knot in the image is linear, whereas the saint Hannes knot is arced. I’m still unsure whether the linearity of knot the knot is necessary, or just an artefact of starting with an equation that depicts a linear (non-arced) helix.
If the case is indeed so that there are two very different regions in the molecule: i.e. the knot(s) and everywhere else, this seems to complicate writing an equation that combines both of these regions seamlessly. While I thought I’d try to come up with an exact equation for the molecular knot, I’m tempted to present this qualitative solution in the knot manuscript instead.
So, I’ll leave this for a while. I’m satisfied that this is qualitatively correct, but quantitatively probably incorrect. I’ll probably find the quantitative solution not by looking at this specific detail, but rather by looking at how this qualitative solution fits a larger whole. This means that I’ll continue to write my knot manuscript. It will probably present me with plenty more problems to be solved.
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