The Borromean rings are a three-component link whose components are unknotted, which is nontrivial although removing any one component leaves an unlink. Its complement has a complete finite-volume hyperbolic three-manifold structure obtained from two regular ideal octahedra.
With group commutator convention , the displayed group presentation describes the fundamental group of the Borromean rings complement. Its abelianization is , since each relator is a group commutator and hence imposes no abelian relation. It embeds discretely in the Gaussian Bianchi group by sending the meridians to , and .
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The Borromean rings are a set of three interlinked rings that are arranged in such a way that no two rings are directly linked together; instead, all three are interlinked with one another as a complete set. The key property of the Borromean rings is that if any one of the rings is removed, the remaining two rings will be unlinked, meaning they will not be entangled with each other.