To me the difference is about things like friction, diameter of the rope, and the physical reasons behind why a knot resists binding, tightening, and so on. The

To me the difference is about things like friction, diameter of the rope, and the physical reasons behind why a knot resists binding, tightening, and so on. The fact that mathematical knots don't have ends isn't that important. If you have two ends of a rope and so long as those ends never move into the knot's "territory", then its dynamic the same as a circle knot. Isn't it?