This is a variation of the question [Can a tangle of arcs interlock?][1], asked by Joseph O'Rourke, and solved. I reproduce the question here:

> Can a (finite) collection of disjoint circle arcs in $\mathbb{R}^3$ be
> interlocked in the sense in that they cannot be separated, i.e. each
> moved arbitrarily far from one another while remaining disjoint (or at
> least never crossing) throughout? (Imagine the arcs are made of rigid
> steel; but infinitely thin.) The arcs may have different radii; each
> spans strictly less than $2 \pi$ in angle, so each has a positive
> "gap" through which arcs may pass.

My proposed variation is:

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

I posted [a comment][2] at the original question, claiming that three circle arcs can be locked. 

![enter image description here][3]

And two cannot

![enter image description here][4]

I soon realized that the examples with three arcs can in fact be unlocked, and I think Joseph O'Rourke [did the same][5].
I reproduce here my solution to unlock them:
![enter image description here][6]

So, the question is still open for two dimensions.


  [1]: http://mathoverflow.net/questions/139105/can-a-tangle-of-arcs-interlock
  [2]: http://mathoverflow.net/a/140555/10095
  [3]: https://i.sstatic.net/j58z0.gif
  [4]: https://i.sstatic.net/gG7Ot.gif
  [5]: http://mathoverflow.net/a/140567/10095
  [6]: https://i.sstatic.net/f5edL.gif