Version 1.0 in Mathematica: https://0x0.st/ip7E.txt Analyzing outputs, I found the following for case (3,4): {X,Y,Z} = {Sin[t] + 2 Sin[3 t], Cos[t] - 2 Cos[3 t], 2 Sin[4 t]} 0 = -81 + 117 X^2 - 40 X^4 + 4 X^6 + 117 Y^2 - 112 X^2 Y^2 + 12 X^4 Y^2 - 40 Y^4 + 12 X^2 Y^4 + 4 Y^6; 0 = -8 X^3 Y + 8 X Y^3 + 27 Z - 24 X^2 Z + 4 X^4 Z - 24 Y^2 Z + 8 X^2 Y^2 Z + 4 Y^4 Z; 0 = 81 - 81 X^2 - 81 Y^2 + 32 X^2 Y^2 + 16 X^2 Z^2 + 16 Y^2 Z^2 Contrary to expectation, we need not two but three constraints. The first two equations define the knot + 8 vertical lines at crossing points. The third constraint filters out vertical lines. --Brad