Can a tangle of arcs interlock?

Question

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:
     
Of course, if one could prove that in any such collection, one arc can be removed to infinity, the result would follow by induction. But an impediment to that approach is that sometimes there is no arc than can be removed while all the others remain fixed.

Another approach would be to reduce the piercing number of the configuration: the number of intersections of an arc with the disks on whose boundary the arcs lie. If the piercing number could always be reduced in any configuration, then it would "only" remain to prove that if there are no disk-arc piercings at all, the configuration can be separated.

Intuitively it seems that no such collection can interlock, but I am not seeing a proof. I'd appreciate any proof ideas—or interlocked configurations!

Answer 1

I believe there is no such locked configuration. The proof is by induction, as you suggest.

Pick any arc and imagine moving it to infinity. Of course, to do this, it will have to pass through some other arcs, and thus this is not a valid motion. We can, however, by picking our motion "generically", ensure that there are just finitely many times when our arc passes through another arc, and that at each of these times, it passes through exactly one other arc at exactly one point. But now if we rotate (in the plane of the circle) the arc during the motion, we can ensure that it's "gap" is moved to each of the points where it used to pass through another arc. Thus we have turned our invalid motion into a valid one.

Answer 2

There is already a beautiful correct answer.

Mine is just an illustrated comment, for the case $\mathbb R^2$, although the question was about $\mathbb R^3$.

Here is how three arcs can be locked, in $\mathbb R^2$:

Here is why two can't be locked, in $\mathbb R^2$:

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Update

Ooops! Actually, both configurations with 3 circles can be unlocked.

So the problem remains open for the $\mathbb R^2$ case too. I made it into a question.

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The question got answered.

Answer 3

@Cristi: Perhaps your first example needs an argument to exclude the following motion, which might only be possible with my generous arc-gap?