Well i am stuck trying to figure out this problem, actulally i see a machine that can make it with a torus, a torus coilling machine. But that not work with a multi turus shape.
I want to automate something like this:
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I've never seen a machine do this, but consider this.
I've recently seen a video in which chain link fence is being manufactured. The individual segments of fence are somewhat "corkscrewed" into the existing assembled fence.
One method I can envision to accomplish your goal is to have a helix (three-dimensional constant diameter spiral) that is threaded into the multi-torus shape until the entire length is captured by the helix. The end of the helix would then capture the thread and unwind itself, pulling the thread to the entry point.
It would be necessary to have a swivel on the end of the helix or to spin the thread source to match the unwinding helix. The former is more likely to snag, while the latter is mechanically more complex.