The DNA strand is a double helix, but it also “supercoils”. Imagine a rope thats stretched taught. Even the straight rope has twisted braids. Thats akin to the DNA double helix. Then imagine twisting the rope until it starts to bunch and twist up on itself. That’s the supercoil. Of course the superstructure of DNA even more complex and dynamic than that

I always wonder why DNA never turns itself into a hard-to-undo knot (which is what invariably happens with headphone wires). Are these twists helping with that?

Part of the answer is the Histones (spools for DNA) https://en.m.wikipedia.org/wiki/Histone

There are other protein complexes that keep the strand in proper shape and form. They unfold the strand periodically, and can cut and repair it as needed.

If my mind serves me correctly, it is part of gene regulation. Genes located in places of the strand that tend to form tight knots are "more difficult" to transcribe, so they express less, to some extent.

The twist probably does induce the kind of tangling that you mentioned. But, there’s a class of proteins called topoisomerases that allow the DNA to unwind itself via some very beautiful mechanisms.

Without directly addressing your question, see https://en.wikipedia.org/wiki/Linking_number and https://en.wikipedia.org/wiki/DNA_supercoil

Note that math knots are always circular, while DNA is often not.