BCC
Continuing in the basic theme of Minimal and Symmetric Minimality, but now with the figure-eight knot on the body-centered cubic lattice rather than the trefoil knot on the simple cubic lattice. To make this, I grabbed the coordinates of a (purportedly?) minimal figure-eight knot on the body-centered cubic lattice from Andrew Rechnitzer's page:
bcc41 = # - Table[Mean[#], {Length[#]}] &@
Import["https://www.math.ubc.ca/~andrewr/knots/data/BCC41.dat"]
After some fooling around to find nice symmetries of this particular embedding of the figure-eight, I came up with the animation, which alternates between viewing along the $x$-axis and along the $z$-axis, as usual using the smoothesttep function for smooth stops and starts.
smootheststep[t_] := -20 t^7 + 70 t^6 - 84 t^5 + 35 t^4;
DynamicModule[{p, q, a, b, n, M, θ, pl,
cols = RGBColor /@ {"#eeeeee", "#222831"}},
Manipulate[
θ = (Pi - ArcCot[Sqrt[1/17 (23 - 16 Sqrt[2])]]) smootheststep[Mod[t, 1]];
Which[t < 1 || t == 2,
{p, q} = {{-1, 0, 0}, Normalize[{0, 1, 1}]};
{n, b} = RotationMatrix[-θ, {-1, -1 - Sqrt[2], 1}].# & /@ {p, q};,
1 <= t < 2,
{p, q} = {{0, 0, 1}, {1, 0, 0}};
{n, b} = RotationMatrix[-θ, {1, -1 - Sqrt[2], 1}].# & /@ {p, q};
];
a = Cross[b, n];
M = {a, b};
pl = M.# & /@ bcc41;
Graphics[{
EdgeForm[Directive[Thickness[.01], JoinForm["Round"], cols[[1]]]],
FaceForm[None], Polygon[pl],
FaceForm[cols[[1]]], Disk[#, .1] & /@ pl},
ImageSize -> 540, PlotRange -> 3, Background -> cols[[-1]]],
{t, 0, 2}]
]
