On Borromean links and related structures

Michael O’Keeffe, Michael M.J. Treacy

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


The creation of knotted, woven and linked molecular structures is an exciting and growing field in synthetic chemistry. Presented here is a description of an extended family of structures related to the classical ‘Borromean rings’, in which no two rings are directly linked. These structures may serve as templates for the designed synthesis of Borromean polycatenanes. Links of n components in which no two are directly linked are termed ‘n-Borromean’ [Liang & Mislow (1994). J. Math. Chem. 16, 27–35]. In the classic Borromean rings the components are three rings (closed loops). More generally, they may be a finite number of periodic objects such as graphs (nets), or sets of strings related by translations as in periodic chain mail. It has been shown [Chamberland & Herman (2015). Math. Intelligencer, 37, 20–25] that the linking patterns can be described by complete directed graphs (known as tournaments) and those up to 13 vertices that are vertex-transitive are enumerated. In turn, these lead to ring-transitive (isonemal) n-Borromean rings. Optimal piecewise-linear embeddings of such structures are given in their highest-symmetry point groups. In particular, isonemal embeddings with rotoinversion symmetry are described for three, five, six, seven, nine, ten, 11, 13 and 14 rings. Piecewise-linear embeddings are also given of isonemal 1- and 2-periodic polycatenanes (chains and chain mail) in their highest-symmetry setting. Also the linking of n-Borromean sets of interleaved honeycomb nets is described.

Original languageEnglish (US)
Pages (from-to)379-391
Number of pages13
JournalActa Crystallographica Section A: Foundations and Advances
Issue numberPart5
StatePublished - 2021


  • Borromean rings
  • Isonemal
  • N-Borromean
  • Piecewise-linear embedding

ASJC Scopus subject areas

  • Structural Biology
  • Biochemistry
  • Materials Science(all)
  • Condensed Matter Physics
  • Physical and Theoretical Chemistry
  • Inorganic Chemistry


Dive into the research topics of 'On Borromean links and related structures'. Together they form a unique fingerprint.

Cite this