For some time I tried to solve the problem of molecular bonding using helical arcs that are as thin as possible, thinking that this allowed me to zoom in on the molecular knot.
I think I thought that at some level of zooming I could ignore the curvature of the arc and concentrate on some other fundamental property of the knot that would emerge when zoomed. However, when I finally realized that the knot emerges from stretching helical arc directly past the center of the knot, was I able to visualize the molecular knot without discontinuities.
So, I decided to scrap the idea of thinning the arc so that I could zoom in on the knot and instead thickened the knot that I could see the whole orbital: not just the knot. After a bit of tweaking, I found the conditions where the above knot is formed. And here it is:
And here’s the knot from a different angle:
Looking at the images as they are, they don’t look very unique. However, you need to show the two basic shapes of Higgs helices with the knot to see what is curious. Here are the just the basic helices without the knot:
And here is the knot and the basic helices on top of each other:
If you look closely, to form the knot you need to stretch the two helices over the knot in a way, where the outer loops are inside the basic helices. This means that the structure would be fine for a single molecule, such as hydrogen (H2), but this once knotted structure can’t form further knots without doubling the number of turns in the structure. I guess it’s not obvious, but both of the two peaks in both of the knotted helices (blue and yellow) are the loop that stretch over the center of the knot. This means that for this kind of a structure to form a supramolecular loop, there needs to be four peaks in both helices.
I know this goes above the heads of most people reading this, but I’m writing this to myself as much as I’m writing this to you, dear reader.
Honestly, just a few days ago I was sure this was too hard a problem and that I should just let it rest, but now it seems blindingly obvious.
However, I still need to confirm the structure with proper equations and show in excel that I can really make this knot. While I might not be ready with that tomorrow, it shouldn’t take too long, either.
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