Classification of links made of rigid circles

Question

I am not very familiar with low-dimensional topology and I was wondering if we know the classification of links (in $\mathbb R^3$) that can be isotoped into a position where every link is a rigid (geometric) circle. In particular,are there hyperbolic links that can be realized in this way ?

Answer 1

It's a great question. But I only know of a few limited negative results.

For example, the Borromean Rings cannot be realized by geometric circles. More generally, any link having the property that any two of its components are unlinked cannot be realized by geometric circles. This was proved by Freedman and Skora; see this wikipedia link for a quick outline of their rather simple proof, and for a reference to their paper (Lemma 3.2 of that paper).

You can say a few more simple things. It's pretty obvious that two geometric circles in $\mathbb R^3$ are either unlinked or they form a Hopf link. So that narrows down the realm of potential positive examples quite a bit, namely those links such that any two of the components are unlinked or form a Hopf link, with at least one pair forming a Hopf link.

Answer 2

One type of these links are known as "great circle links." Usually knot theory is done in the 3-sphere rather than $\mathbb{R}^3$, but one can go back and forth between these perspectives using a stereographic projection with respect to a point away from the knot/link. Stereographic projection sends geodesics that don't pass through the projection point to circles, so great circle links give one class of links made from rigid circles.

Genevieve Walsh has a thesis that classifies great circle links up to five components. She also has a paper about great circle links.

It seems unlikely that all rigid circle links in $\mathbb{R}^3$ come from stereographic projections of great circle links, but I don't have any good reason to think it's impossible.