Minimal

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

Source code and further explanation: https://community.wolfram.com/groups/-/m/t/1634541

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@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:

@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 (https://doi.org/10.1088/1751-8113/42/47/475006), 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.

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

0xDE@11011110@mathstodon.xyz@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?