Uncaught PythonError: Traceback (most recent call last):
  File "<exec>", line 22, in resetKnot
  File "<exec>", line 473, in generateKnot
  File "/lib/python3.12/site-packages/PIL/Image.py", line 2941, in new
    return im._new(core.fill(mode, size, color))
                   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

1) Draw the 'skeleton' as a connected (simple?) graph in the plane

2) Place crosses at the midpoint of each edge

3) Connect the crosses with shortest (non-crossing!) connections (bit vague this, but is more obvious by hand)

4) Erase the crosses, and run over the line, assigning under/over as appropriate - you can also thicken at this step

This gives good free-standing knots, although may be more work for the dense knotwork in the OP's examples.