I think this is a genuine math problem. And it's somehow related to knot energy but not directly solved by the latter.
Why can't I tie a hard knot on a rope of infinite length?
By infinity I mean you never pull back either ends of the rope.
And I attentively add a constraint to the rope that the rope is smooth thus no friction. So the problem is only about the topology (not exactly) of that rope.
I cannot think of a better word than topology since the topology of the rope should never change. But apparently I don't know a better one.
In my preliminary thoughts, there should be a solution that performs a integration involving the direction and twist of the rope along the infinite length. And the sign of the integration determines whether it's a hard knot or not. And the fact that you cannot pull back the end means you cannot change that sign.