TITLE: Are there two deformations of a wire loop such that neither can pass through the other?
QUESTION [16 upvotes]: My 10-year old son asked me this simple question, and I've been unable to answer it.
Suppose we start with two (unlinked) circular wire loops (maybe different sizes), and allow both to be continuously deformed in three-dimensional space, the only constraint being that different parts of the loop mustn't touch. The two loops can be deformed in different ways.
Is it possible to do so in such a way that neither of the loops (now considered rigid) can be passed through the other by rigid motion? Consider the wire to be one-dimensional; the definition of 'passing through' permits the two loops to touch (as is obviously necessary in the case of two identical circles), but not to cross each other.
For a formal definition of 'passing through', suppose the two loops are A (fixed) and B (moving). If there is a point on B that traces a path C through space as we move B, such that C is a closed loop that is linked with A, then we say B has passed through A.
Edit Joseph O'Rourke has given an interesting example where the wires are 'interlocked'. So I'd now like to add a condition to exclude this situation, and say that, prior to the 'passing through', the wires must be separated (occupying different sides of some plane).
Edit Experimenting with garden wire, inspired by some of the comments below, I came up with a configuration that may work:

But even with this I am not certain. Ideally I'd like a simple provable example. In particular, could there be an example where both loops are planar?

REPLY [8 votes]: I think there are quite a few examples.   Here is one that is the tip of a family of ideas.  
Let's call the two wire loops $L_1$ and $L_2$. 
I'll start by describing a slightly more pliable definition of what it means for one loop to pass through another.  This can be done without too much technical detail, as we are moving our loops with rigid motions.
Fix an embedded spanning disc $D_1$ for $L_1$ and $D_2$ for $L_2$.  As you move $L_1$ and $L_2$ by rigid motions, imagine doing the same rigid motion to $D_1$ and $D_2$. 
The loop $L_1$ passes through $L_2$ (once) if, in the family of loops, any point on the spanning disc $D_1$ intersects the spanning disc $D_2$ transversely exactly once (through the motion). To make this definition fully general you'd need to bring in some transversality theory, but let's ignore that detail for now.
Example. 
$L_1$ is a circle of radius 1, and $L_2$ is a "double wound" helix about another circle of radius 1, as in the picture.  Basically this is a gimmick to make the loop $L_2$ behave as if it were a "thick" circle. 

If you think about the rigid motions that put the loop $L_1$ so that its centre is on the spanning disc $D_2$, they either force $L_1$ and $L_2$ to intersect, or they link. 
I hope that makes some sense.
This example makes me think you should also be able to come up with examples of loops $L_1$ and $L_2$ where $L_1$ can't pass through $L_2$, but $L_2$ can pass through $L_1$ $n$ times for some $n>1$, but not with $n=1$.  i.e. to have one loop pass through another, it might have to do it multiple times, but it won't be able to do it just a single time.