This is a variation of the question Can a tangle of arcs interlock?, asked by Joseph O'Rourke, and solved.

My proposed variation is:

Can they interlock in $\mathbb R^2$?

I posted a comment at the original question, claiming that three circle arcs can be locked.

And two cannot

I soon realized that the examples with three arcs can in fact be unlocked, and I think Joseph O'Rourke did the same. I reproduce here my solution to unlock them:

So, the question is still open for two dimensions.