# Can a tangle of arcs interlock in plane?

This is a variation of the question Can a tangle of arcs interlock?, 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 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.

## 1 Answer

Surely if you take your first example, and put two more green arcs, one in the red circle but not in the blue, and one in the blue circle but not in the red, about as large as they possibly can be, that will hold the configuration fairly rigid and prevent that unlinking operation?

I don't see that anything significant can be done here except idly rotating the green circles in place.

• Very nice and quick! – Cristi Stoica Aug 27 '13 at 15:47
• In fact I think it'll suffice to have only one of the two: hence I think a set of four arcs need not be unlinkable. – James Cranch Aug 27 '13 at 15:51
• Would be interesting if there is a critical value for the gaps, that decides about the existence of interlocking ensembles of circular arcs. – Manfred Weis Aug 28 '13 at 19:35