Brunnian link

Brunn's 1892 article Über Verkettung included examples of such links.

[2] In 2020, new and much more complicated Brunnian links were discovered in [3] using highly flexible geometric-topology methods.

The proof, by Michael Freedman and Richard Skora, embeds the three-dimensional space containing the link as the boundary of a Poincaré ball model of four-dimensional hyperbolic space, and considers the hyperbolic convex hulls of the circles.

Milnor showed that the group elements that do correspond to Brunnian links are related to the graded Lie algebra of the lower central series of the free group, which can be interpreted as "relations" in the free Lie algebra.

[5] More interestingly, a canonical geometric decomposition in terms of satellite-sum and satellite-tie, which is simpler than JSJ-decomposition, for Brunnian links, was developed.

Many disentanglement puzzles and some mechanical puzzles are variants of Brunnian Links, with the goal being to free a single piece only partially linked to the rest, thus dismantling the structure.

This four-component link is a Brunnian link.
The Borromean rings are the simplest Brunnian link.
Six-component "rubberband" Brunnian link. The same construction leads to Brunnian links with arbitrary numbers of components.
The standard braid is Brunnian: if one removes the black strand, the blue strand is always on top of the red strand, and they are thus not braided around each other; likewise for removing other strands.
Rainbow loom bracelet showing Brunnian chains