Wonderful work! So newcomers to knot theory can appreciate deeper your art here are some basic visuals of what $7_7$ knot is:

KnotData[{7, 7}, #] & /@ {"Image", "KnotDiagram"}

Looks nice!

Just to be ornery, I came up with a variation that combines my "rational smoothstep" with spherical linear interpolation ("slerp"):

DynamicModule[{b, n, u, ? = N[2 ?/3] , cols = RGBColor /@ {"#621295", "#e3e7f1"}},
              Manipulate[u = (#^3/(1 - 3 # (1 - #))) &[Mod[t, 1]];
                        {n, b} = Which[t < 1 || t == 2,
                                       {{-1, 0, 0}, {0, 0, 1}} Sin[? (1 - u)] +
                                       {{0, 0, 1}, {0, -1, 0}} Sin[? u],
                                       1 ? t < 2,
                                       {{0, 0, 1}, {0, -1, 0}} Sin[? (1 - u)] +
                                       {{1, 0, 0}, {0, 0, 1}} Sin[? u]]/Sin[?];
                        Graphics[{cols[[1]],
                                  Polygon[bcc77.Transpose[{Cross[b, n], b}]]},
                                  ImageSize -> 540, PlotRange -> 4,
                                  Background -> cols[[-1]]], {t, 0, 2}]]