[a333999]

> any description purely in terms of pairwise relationships would miss the fact that the three rings are connected.

Here is one:

A is connected to B

B is connected to C

C is connected to A

This is a description in terms of pairs, and from it is trivial to deduce how the rings are connected.

[defrost]

But they're not pairwise connected.

In the text:

   If you only consider two of the rings and ignore the third, then any pair can be smoothly separated.

and from looking at the diagram. Carefully looking.

[abelaer]

The pairwise description you list correspond to a different link, namely, one where each pair is actually connected. For the shown rings, the 'correct' description would be:

A is not connected to B B is not connected to C C is not connected to A A, B, and C are connected

This seems paradoxical, but the paradox is resolved by the 'higher-order' linkage.