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Minimal

The shortest-possible trefoil knot on the simple cubic lattice.

Source code and further explanation: community.wolfram.com/groups/-

@shonk Nice, but it's too bad the positions that it stops long enough to let you see it clearly are all positions where the projection maps more than one lattice point to the same point of the plane, so that the height of the knot curve above the projection plane is ambiguous.

@11011110 Well, that was kind of intentional (and also, being able to see the intermediate frames more clearly wouldn't have helped, because I cheated and just showed projections to the plane with no crossing information).

But, just for you, here are (slightly low res) 3D views of the midpoint between each pause in the animation:

@shonk Thanks. I don't have enough zomes on hand right now to tell for sure, but it seems there's a much more symmetric and equally-minimal-length embedding, with segments of lengths \((3,2,1,2)^3\) and sixfold (3-dihedral) symmetry. But maybe you were looking for a smaller bounding box rather than more symmetry?

@11011110 It’s entirely likely. I just took the one that’s built into KnotPlot, but there are 3496 minimal lattice trefoils (doi.org/10.1007/BF02188227), so I wouldn’t be at all surprised if one is substantially more symmetric. Let me know if you figure it out!

@shonk I got it to work in vzome (vzome.com/home/) instead of waiting until tomorrow when I can find more of the physical ones at my office. See attached image.

@11011110 Nice! That totally works! It’s in Diao’s class (c), but surprisingly it’s not the one he shows. It also looks to me like Scharein et al.’s 2–17 (doi.org/10.1088/1751-8113/42/4), but they don’t remark on it either.

I wonder whether one should expect highly symmetric minimal lattice knots in general; my guess is probably not, but all the evidence seems to suggest that there are lots of minimal examples for each knot type.

@shonk Well, you're not going to get symmetries that can't be realized on lattices, like for the (5,2)-torus knot. But maybe it's an interesting question whether every lattice-realizable symmetry of the knot can be realized by a minimal lattice knot.

@11011110 Yeah, that does seem like the right question, doesn’t it?

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