How do I wrap a path around a part of another path?

I want to weave a line (using a brush that looks like a chain) around another path but I don't know how to move just portions of the line below the path while keeping other portions above it.

Here is the step-by-step of the Live Paint method.
Two simple paths, before wrapping:
Here the eyedropper has been clicked on the pink shape, so that the formerly-gold areas selected above have been filled with pink. Also the stroke styles of the overlapping pieces have been changed as well. (You may need to change the options of the Live Paint Select tool to enable it to select strokes.)
The neat thing about the Live Paint method is that you have only two paths, and any changes made to them intelligently repaint the overlapping regions. Here we see what happens after you rotate the gold path:
But paths in Live Paint groups cannot carry brushes or effects. So for complex designs the Clipping Mask method is still best, even though it means that if you want to edit the duplicated shape you will have to either redo the duplication, or else make exactly the same edits to both copies. So that can be another factor to take into account when deciding which of the two designs to duplicate: if you expect to be more likely to want to edit one of them than the other, then the one you want to edit should be the one that isn't duplicated.

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