For people who don't want to download software just to get the needed points for the knot:

lattice31 = AnglePath3D[{19/24, 4/3, 11/8},
                        {{0, Pi/2}, {0, 0}, {-Pi/2, 0}, {0, Pi/2}, {0, 0}, {0, -Pi/2},
                         {-Pi/2, 0}, {Pi/2, 0}, {0, -Pi/2}, {0, 0}, {0, -Pi/2}, {0, 0},
                         {0, -Pi/2}, {0, Pi/2}, {Pi/2, 0}, {0, 0}, {Pi/2, 0}, {0, 0},
                         {Pi/2, 0}, {0, 0}, {0, 0}, {0, Pi/2}, {Pi/2, 0}}];

As for the main code itself: the standard advice of replacing M.# & /@ pts with pts.Transpose[M] and RotationMatrix[-θ, axis].# & /@ {p, q} with {p, q}.RotationMatrix[θ, axis] applies. ;)

For better or for worse, Mathematica's matrices (or list of lists, however your preference) are row-oriented. I admit that it also took me some time to adjust my conventional mental images of linear algebra to adapt them to programming in Mathematica.