Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are linked but where removing any one ring leaves the other two unconnected. In other words, no two of the three rings are linked with each other as a Hopf link, but nonetheless all three are linked. The Borromean rings are one of a class of such links called Brunnian links.

Borromean rings
L6a4
Braid length6
Braid no.3
Crossing no.6
Hyperbolic volume7.327724753
Stick no.9
Unknotting no.2
Conway notation[.1]
A-B notation63
2
ThistlethwaiteL6a4
Last /NextL6a3 / L6a5
Other
alternating, hyperbolic

Mathematical properties

Impossibility of perfectly circular rings

Although the typical picture of the Borromean rings (above right picture) may lead one to think the link can be formed from geometrically ideal circular rings, they cannot be. Michael H. Freedman and Richard Skora (1987) proved that a certain class of links, including the Borromean links, cannot be exactly circular. Alternatively, this can be seen from considering the link diagram: if one assumes that circles 1 and 2 touch at their two crossing points, then they lie in either a plane or a sphere. In either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible; see Bernt Lindström and Hans-Olov Zetterström (1991).

A realization of the Borromean rings as ellipses
3D image of Borromean Rings

However, one can use ellipses (right picture). These may be taken to be of arbitrarily small eccentricity; i.e. no matter how close to being circular their shape may be, as long as they are not perfectly circular, they can form Borromean links if suitably positioned; as an example, thin circles made from bendable elastic wire may be used as Borromean rings.

Relationship to the octahedral graph

Apart from indicating which strand crosses over the other, link diagrams use the same notation to show two strands crossing, as graph diagrams use to show four edges meeting at a common vertex. Accordingly, the graph of the regular octahedron may be converted into a link diagram by prescribing that, as a strand follows successive edges, it alternates between passing over a vertex and passing under the next. The result has three separate loops, linked together as Borromean rings.[1]

Linking
Further information: Link group

In knot theory, the Borromean rings are a simple example of a Brunnian link: although each pair of rings is unlinked, the whole link cannot be unlinked. There are a number of ways of seeing this.

Simplest is that the fundamental group of the complement of two unlinked circles is the free group on two generators, a and b, by the Seifert–van Kampen theorem, and then the third loop has the class of the commutator, , as one can see from the link diagram: over one, over the next, back under the first, back under the second. This is non-trivial in the fundamental group, and thus the Borromean rings are linked.

Another way is that the cohomology of the complement supports a non-trivial Massey product, which is not the case for the unlink. This is a simple example of the Massey product and further, the algebra corresponds to the geometry: a 3-fold Massey product is a 3-fold product which is only defined if all the 2-fold products vanish, which corresponds to the Borromean rings being pairwise unlinked (2-fold products vanish), but linked overall (3-fold product does not vanish).

In arithmetic topology, there is an analogy between knots and prime numbers in which one considers links between primes. The triple of primes (13, 61, 937) are linked modulo 2 (the Rédei symbol is −1) but are pairwise unlinked modulo 2 (the Legendre symbols are all 1). Therefore, these primes have been called a "proper Borromean triple modulo 2"[2] or "mod 2 Borromean primes".[3]

Hyperbolic geometry

The Borromean rings are a hyperbolic link: the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume. The canonical (Epstein–Penner) polyhedral decomposition of the complement consists of two regular ideal octahedra. The volume is 16Л(π/4) = 7.32772… where Л is the Lobachevsky function.[4]

Name and history
Valknut on Stora Hammars I stone

The name "Borromean rings" comes from their use in the coat of arms of the aristocratic Borromeo family in Northern Italy.[5] The link itself is much older and has appeared in the form of the valknut on Norse image stones dating back to the 7th century.[6]

The Borromean rings as a symbol of the Christian Trinity, from a 13th-century manuscript.

The Borromean rings have been used in different contexts to indicate strength in unity, e.g., in religion or art. In particular, some have used the design to symbolize the Trinity. The psychoanalyst Jacques Lacan found inspiration in the Borromean rings as a model for his topology of human subjectivity, with each ring representing a fundamental Lacanian component of reality (the "real", the "imaginary", and the "symbolic").[7]

The rings were used as the logo of Ballantine beer, and are still used by the Ballantine brand beer, now distributed by the current brand owner, the Pabst Brewing Company.[8]

The Borromean rings, especially their mathematical properties, were featured by Martin Gardner in his September 1961 "Mathematical Games column" in Scientific American.

In 2006, the International Mathematical Union decided at the 25th International Congress of Mathematicians in Madrid, Spain to use a new logo based on the Borromean rings.[9]

A stone pillar at Marundeeswarar Temple in Thiruvanmiyur, Chennai, Tamil Nadu, India, has such a figure dating to before the 6th century.[10][11]

Partial and multiple rings
A monkey's fist knot

In medieval and renaissance Europe, a number of visual signs are found that consist of three elements interlaced together in the same way that the Borromean rings are shown interlaced (in their conventional two-dimensional depiction), but the individual elements are not closed loops. Examples of such symbols are the Snoldelev stone horns and the Diana of Poitiers crescents. An example with three distinct elements is the logo of Sport Club Internacional. Less-related visual signs include the Gankyil and the Venn diagram on three sets.

Similarly, a monkey's fist knot is essentially a 3-dimensional representation of the Borromean rings, albeit with three layers, in most cases.

The Discordian "mandala", containing five Borromean rings configurations

Using the pattern in the incomplete Borromean rings, one can balance three knives on three supports, such as three bottles or glasses, providing a support in the middle for a fourth bottle or glass.[12]

Some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol in Discordianism, based on a depiction in the Principia Discordia.

Connection with braids
The standard 3-strand braid corresponds to the Borromean rings.

If one cuts the Borromean rings, one obtains one iteration of the standard braid; conversely, if one ties together the ends of (one iteration of) a standard braid, one obtains the Borromean rings. Just as removing one Borromean ring unlinks the remaining two, removing one strand of the standard braid unbraids the other two: they are the basic Brunnian link and Brunnian braid, respectively.

In the standard link diagram, the Borromean rings are ordered non-transitively, in a cyclic order. Using the colors above, these are red over green, green over blue, blue over red – and thus after removing any one ring, for the remaining two, one is above the other and they can be unlinked. Similarly, in the standard braid, each strand is above one of the others and below the other.

Realizations
Crystal structure of molecular Borromean rings reported by Stoddart et al. (Science 2004)[13]

Molecular Borromean rings are the molecular counterparts of Borromean rings, which are mechanically-interlocked molecular architectures. In 1997, biologists Chengde Mao and coworkers of New York University succeeded in constructing a set of rings from DNA.[14] In 2003, chemist Fraser Stoddart and coworkers at UCLA utilised coordination chemistry to construct a set of rings in one step from 18 components.[13] Borromean ring structures have been shown to be an effective way to represent the structure of certain atomically precise noble metal clusters which are shielded by a surface layer of thiolate ligands (-SR), such as Au25(SR)18 and Ag25(SR)18.[15] A library of Borromean networks has been synthesized by design by Giuseppe Resnati and coworkers via halogen bond driven self-assembly.[16] In order to access the molecular Borromean ring consisting of three unequal cycles a step-by-step synthesis was proposed by Jay S. Siegel and coworkers.[17]

A quantum-mechanical analog of Borromean rings is called a halo state or an Efimov state (the existence of such states was predicted by physicist Vitaly Efimov, in 1970). For the first time the research group of Rudolf Grimm and Hanns-Christoph Nägerl from the Institute for Experimental Physics (University of Innsbruck, Austria) experimentally confirmed such a state in an ultracold gas of caesium atoms in 2006, and published their findings in the scientific journal Nature.[18] A team of physicists led by Randall Hulet of Rice University in Houston achieved this with a set of three bound lithium atoms and published their findings in the online journal Science Express.[19] In 2010, a team led by K. Tanaka created an Efimov state within a nucleus.[20]

See also

Notes

    References
    1. N. L. Biggs (1 March 1981). "T. P. Kirkman, Mathematician". Bull. London Math. Soc. 13 (2): 116. doi:10.1112/blms/13.2.97. Retrieved 24 January 2017.
    2. Vogel, Denis (2005), Masseyprodukte in der Galoiskohomologie von Zahlkörpern [Massey products in the Galois cohomology of number fields], Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005, Göttingen: Universitätsdrucke Göttingen, pp. 93–98, doi:10.11588/heidok.00004418, MR 2206880
    3. Morishita, Masanori (22 April 2009), Analogies between Knots and Primes, 3-Manifolds and Number Rings, arXiv:0904.3399, Bibcode:2009arXiv0904.3399M
    4. William Thurston (March 2002), "7. Computation of volume" (PDF), The Geometry and Topology of Three-Manifolds, p. 165
    5. Schoeck, Richard J. (Spring 1968), "Mathematics and the languages of literary criticism", The Journal of Aesthetics and Art Criticism, 26 (3): 367–376, doi:10.2307/429121, JSTOR 429121
    6. Bruns, Carson J.; Stoddart, J. Fraser (2011), "The mechanical bond: A work of art", in Fabbrizzi, L. (ed.), Beauty in Chemistry, Topics in Current Chemistry, 323, Springer-Verlag, pp. 19–72, doi:10.1007/128_2011_296
    7. Ragland-Sullivan, Ellie; Milovanovic, Dragan (2004), "Introduction: Topologically Speaking", Lacan: Topologically Speaking, Other Press, ISBN 9781892746764
    8. Glick, Ned (September 1999), "The 3-ring symbol of Ballantine Beer", The mathematical tourist, The Mathematical Intelligencer, 21 (4): 15–16, doi:10.1007/bf03025332
    9. ICM 2006
    10. Arul Lakshminarayan (May 2007). "Borromean Triangles and Prime Knots in an Ancient Temple" (PDF). Indian Academy of Sciences. Retrieved 18 September 2014.
    11. Blog entry by Arul Lakshminarayan
    12. Comments on Knives And Beer Bar Trick: Amazing Balance
    13. Kelly S. Chichak; Stuart J. Cantrill; Anthony R. Pease; Sheng-Hsien Chiu; Gareth W. V. Cave; Jerry L. Atwood; J. Fraser Stoddart (28 May 2004). "Molecular Borromean Rings" (PDF). Science. 304 (5675): 1308–1312. Bibcode:2004Sci...304.1308C. doi:10.1126/science.1096914. PMID 15166376.
    14. C. Mao; W. Sun; N. C. Seeman (1997). "Assembly of Borromean rings from DNA". Nature. 386 (6621): 137–138. Bibcode:1997Natur.386..137M. doi:10.1038/386137b0. PMID 9062186.
    15. Natarajan, Ganapati; Mathew, Ammu; Negishi, Yuichi; Whetten, Robert L.; Pradeep, Thalappil (2015-12-02). "A Unified Framework for Understanding the Structure and Modifications of Atomically Precise Monolayer Protected Gold Clusters". The Journal of Physical Chemistry C. 119 (49): 27768–27785. doi:10.1021/acs.jpcc.5b08193. ISSN 1932-7447.
    16. Vijith Kumar; Tullio Pilati; Giancarlo Terraneo; Franck Meyer; Pierangelo Metrangolo; Giuseppe Resnati (2017). "Halogen bonded Borromean networks by design: topology invariance and metric tuning in a library of multi-component systems". Chemical Science. 8 (3): 1801–1810. doi:10.1039/C6SC04478F. PMC 5477818. PMID 28694953.
    17. Veliks, Janis; Seifert, Helen M.; Frantz, Derik K.; Klosterman, Jeremy K.; Tseng, Jui-Chang; Linden, Anthony; Siegel, Jay S. (2016). "Towards the molecular Borromean link with three unequal rings: double-threaded ruthenium(ii) ring-in-ring complexes". Organic Chemistry Frontiers. 3 (6): 667–672. doi:10.1039/c6qo00025h.
    18. T. Kraemer; M. Mark; P. Waldburger; J. G. Danzl; C. Chin; B. Engeser; A. D. Lange; K. Pilch; A. Jaakkola; H.-C. Nägerl; R. Grimm (2006). "Evidence for Efimov quantum states in an ultracold gas of caesium atoms". Nature. 440 (7082): 315–318. arXiv:cond-mat/0512394. Bibcode:2006Natur.440..315K. doi:10.1038/nature04626. PMID 16541068.
    19. Clara Moskowitz (December 16, 2009), Strange Physical Theory Proved After Nearly 40 Years, Live Science
    20. K. Tanaka (2010), "Observation of a Large Reaction Cross Section in the Drip-Line Nucleus 22C", Physical Review Letters, 104 (6): 062701, Bibcode:2010PhRvL.104f2701T, doi:10.1103/PhysRevLett.104.062701, PMID 20366816

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