Borromean Rings Question

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A link of three components in $\mathbb R^3$ is the disjoint union $L=C_1\cup C_2\cup C_3$ of three embedded closed curves. Each $C_i$ is unknotted if it is ambient isotopic to a round circle. A two-component unlink is ambient isotopic to two separated circles. The Borromean link type is the standard three-component link for which deleting any component leaves a two-component unlink, while all three components together are not an unlink; specifically the conventional Borromean link, rather than an arbitrary Brunnian link, is intended.

Statement recorded in the supplied source.

Do there exist three pairwise disjoint smooth unknotted closed curves $C_1,C_2,C_3\subset\mathbb R^3$, not all geometrically round circles, whose union has the Borromean link type? [S1][S2] The literal existence question has affirmative examples: the rectangular model in the cited source can have its corners smoothed while preserving its link type.

Short English statement

Can three individually unknotted loops, with noncircular shapes permitted, form the standard Borromean link?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1385 (GEOM-026). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Dave Auckly, Folklore, the Borromean rings, the icosahedron, and three dimensions (2020), arXiv:2009.09836v1, "The Borromean rings", pp. 3--7, Figures 2--5. https://arxiv.org/abs/2009.09836v1.