It's very hard to point you in the right direction without actually seeing the artwork. If you want a rope to start at the edge of the frame and animate in and tie it's self in a knot then you'll probably have to use the same technique that a cell animator would use - create every frame. The only other option I can think of is to use a 3D application. Blender is plenty capable and it's open source and free. http://blender.org
Anything you do entirely in After Effects that involves a bending and moving rope is going to require a lot of artwork. If you just want to reveal the rope already tied in a knot then the easiest way to do so is to use a shape layer with a stroke and a trim path animator as a track matte or set matte and the artwork broken into layers where the rope passes in front of itself. A knot would require at least 3 layers to get the effect of the rope tying it's self in a knot.
This is what you are going to need:
AE itself has no path distortion effect. Other than that of course there's all kinds of manual animation techniques using masks and manual transformations. Either way, even with the plug-in - should you decide to buy it - it's not going to be pushbotton animation. You have to spend some time on it.
Cohen: It looks like this post is a parallel topic to the one you posted here: Creating a Wood Burned Effect with Text and Logo
It helps to show the artwork you have in mind. Please confirm if this is the same project.
Yes, sorry. These are linked. I was thinking of a way to reach more people.
The art in the other post correlates with the question in this post.