Your intuition is correct, it doesn't! A "3D hyperrope" is in fact just the surface of a ball[1], and it turns out that you can actually form non-trivial knots of that spherical surface in a 4-dimensional ambient space (and analogously they can be un-knotted if you then move up to 5-dimension ambient space, although the mechanics for doing so might be a little trickier than in the 1d-in-4d case). In fact, if you have a k-dimensional sphere, you can always knot it up in a k+2 dimensional ambient space (and can then always be unknotted if you add enough additional dimensions).
[1] note that a [loop of] rope is actually a 1-dimensional object (it only has length, no width), so the next dimension up should be a 2-dimensional object, which is true of the surface of a ball. a topologist would call these things a 1-sphere and a 2-sphere, respectively
Any time I am tempted to feel smart, I try to go and study some linear algebra and walk away humbled. I will be spending 20-30 minutes probably trying to understand what you said (and I think you typed it out quite reasonably), but first I have to figure out how... a 3D hyperrope is the same as a surface of a ball...
[1] note that a [loop of] rope is actually a 1-dimensional object (it only has length, no width), so the next dimension up should be a 2-dimensional object, which is true of the surface of a ball. a topologist would call these things a 1-sphere and a 2-sphere, respectively