Forming the Borromean Rings out of arbitrary polygonal unknots

TL;DR

This paper proves that any three polygonal unknots in three-dimensional space can be arranged to form the Borromean rings using rigid motions and scaling, with special cases requiring no scaling.

Contribution

It establishes the possibility of forming Borromean rings from arbitrary polygonal unknots, including near-circular shapes, expanding understanding of knot configurations.

Findings

Any three polygonal unknots can form Borromean rings with rigid motions and scaling.

If two unknots are planar, no scaling is needed.

Near-circular unknots can still form Borromean rings despite their similarity to circles.

Abstract

We prove the perhaps surprising result that given any three polygonal unknots in $\R^3$, then we may form the Borromean rings out of them through rigid motions of $\R^3$ applied to the individual components together with possible scaling of the components. We also prove that if at least two of the unknots are planar, then we do not need scaling. This is true even for a set of three polygonal unknots that are arbitrarily close to three circles, which themselves cannot be usedfigure to form the Borromean Rings.