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Kollros' Theorem, with extended turns allowed, says

For every ring containing $p$ spheres, there exists a ring of $q$ spheres, each touching each of the $p$ spheres [...] if more than one turn is allowed, then $$(p-2r)(q-2s)=4rs\quad\quad\left[\text{ or }\frac{r}{p}+\frac{s}{q}=\frac{1}{2},\text{ -anon }\right]$$ where $r$ and $s$ are the numbers of turns on both necklaces before closing.

(The first sentence looks like it contains an error: there shouldn't be a $q$ for every $p$ - most likely it should read that two interlocking rings of spheres correspond to solutions to the Diophantine equation and vice-versa.)

Firstly, in terms of tangencies and intersections, what precisely is a turn? Secondly, can this be seen via inversive geometry in an analogous way to Soddy's hexlet ($r=s=1$, $p=3$, $q=6$)? Inversion preserves tangencies, and sends spheres intersecting the center of inversion to planes, so for Soddy's hexlet we can invert about any tangent point of the first ring of $3$ spheres so that we get two parallel planes and a sphere stuck between them, reducing the problem of finding the interlocking ring of spheres to the annular case, which is equivalent to working with a Steiner chain (or more specifically the kissing number problem), and the $p=4$, $q=4$ case can be reduced similarly.

$\hskip 0.2in$ Soddy's hexlet annular hexlet

There doesn't appear to be any reference for this, as MathWorld cites a "personal communication."

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