Inmathematics, theBorromean rings^[a] are threesimple closed curves in three-dimensional space that aretopologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of aVenn diagram,alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing.

Borromean rings
L6a4
Crossing no.	6
Hyperbolic volume	7.327724753
Stick no.	9
Conway notation	.1
A–B notation	63 2
Thistlethwaite	L6a4
Other
alternating,hyperbolic

The Borromean rings are named after the ItalianHouse of Borromeo, who used the circular form of these rings as an element of theircoat of arms, but designs based on the Borromean rings have been used in many cultures, including by theNorsemen and in Japan. They have been used in Christian symbolism as a sign of theTrinity, and in modern commerce as the logo ofBallantine beer, giving them the alternative nameBallantine rings. Physical instances of the Borromean rings have been made from linkedDNA or other molecules, and they have analogues in theEfimov state andBorromean nuclei, both of which have three components bound to each other although no two of them are bound.

Geometrically, the Borromean rings may be realized by linkedellipses, or (using the vertices of a regularicosahedron) by linkedgolden rectangles. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. Inknot theory, the Borromean rings can be proved to be linked by counting theirFoxn-colorings. As links, they areBrunnian,alternating,algebraic, andhyperbolic. Inarithmetic topology, certain triples ofprime numbers have analogous linking properties to the Borromean rings.

It is common in mathematics publications that define the Borromean rings to do so as alink diagram, a drawing of curves in the plane with crossings marked to indicate which curve or part of a curve passes above or below at each crossing. Such a drawing can be transformed into a system of curves in three-dimensional space by embedding the plane into space and deforming the curves drawn on it above or below the embedded plane at each crossing, as indicated in the diagram. The commonly-used diagram for the Borromean rings consists of three equalcircles centered at the points of anequilateral triangle, close enough together that their interiors have a common intersection (such as in aVenn diagram or the three circles used to define theReuleaux triangle). Its crossingsalternate between above and below when considered in consecutive order around each circle;^[2][3][4] another equivalent way to describe the over-under relation between the three circles is that each circle passes over a second circle at both of their crossings, and under the third circle at both of their crossings.^[5] Two links are said to be equivalent if there is a continuous deformation of space (anambient isotopy) taking one to another, and the Borromean rings may refer to any link that is equivalent in this sense to the standard diagram for this link.^[4]

InThe Knot Atlas, the Borromean rings are denoted with the code "L6a4"; the notation means that this is a link with six crossings and an alternating diagram, the fourth of five alternating 6-crossing links identified byMorwen Thistlethwaite in a list of allprime links with up to 13 crossings.^[6] In the tables of knots and links in Dale Rolfsen's 1976 bookKnots and Links, extending earlier listings in the 1920s by Alexander and Briggs, the Borromean rings were given theAlexander–Briggs notation "6³
₂", meaning that this is the second of three 6-crossing 3-component links to be listed.^[6][7] TheConway notation for the Borromean rings, ".1", is an abbreviated description of the standard link diagram for this link.^[8]

The name "Borromean rings" comes from the use of these rings, in the form of three linked circles, in thecoat of arms of thearistocraticBorromeo family inNorthern Italy.^[9][10] The link itself is much older and has appeared in the form of thevalknut, three linkedequilateral triangles with parallel sides, onNorseimage stones dating back to the 7th century.^[11] TheŌmiwa Shrine in Japan is also decorated with a motif of the Borromean rings, in their conventional circular form.^[2] A stone pillar in the 6th-centuryMarundeeswarar Temple in India shows three equilateral triangles rotated from each other to form a regularenneagram; like the Borromean rings these three triangles are linked and not pairwise linked,^[12] but this crossing pattern describes a different link than the Borromean rings.^[13]

The Borromean rings have been used in different contexts to indicate strength in unity.^[14] In particular, some have used the design to symbolize theTrinity.^[3] A 13th-century French manuscript depicting the Borromean rings labeled as unity in trinity was lost in a fire in the 1940s, but reproduced in an 1843 book byAdolphe Napoléon Didron. Didron and others have speculated that the description of the Trinity as three equal circles in canto 33 ofDante'sParadiso was inspired by similar images, although Dante does not detail the geometric arrangement of these circles.^[15][16] The psychoanalystJacques 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").^[17]

The rings were used as the logo ofBallantine beer, and are still used by the Ballantine brand beer, now distributed by the current brand owner, thePabst Brewing Company.^[18][19] For this reason they have sometimes been called the "Ballantine rings".^[3][18]

The first work ofknot theory to include the Borromean rings was a catalog of knots and links compiled in 1876 byPeter Tait.^[3] Inrecreational mathematics, the Borromean rings were popularized byMartin Gardner, who featuredSeifert surfaces for the Borromean rings in his September 1961 "Mathematical Games" column inScientific American.^[19] In 2006, theInternational Mathematical Union decided at the25th International Congress of Mathematicians in Madrid, Spain to use a new logo based on the Borromean rings.^[2]

In medieval and renaissance Europe, a number of visual signs consist of three elements interlaced together in the same way that the Borromean rings are shown interlaced (in their conventional two-dimensional depiction), but with individual elements that are not closed loops. Examples of such symbols are theSnoldelev stone horns^[20] and theDiana of Poitiers crescents.^[3]

Some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol inDiscordianism, based on a depiction in thePrincipia Discordia.^[21]

Inknot theory, the Borromean rings are a simple example of aBrunnian link, a link that cannot be separated but that falls apart into separate unknotted loops as soon as any one of its components is removed. There are infinitely many Brunnian links, and infinitely many three-curve Brunnian links, of which the Borromean rings are the simplest.^[13][22]

There are a number of ways of seeing that the Borromean rings are linked. One is to useFoxn-colorings, colorings of the arcs of a link diagram with the integersmodulon so that at each crossing, the two colors at the undercrossing have the same average (modulon) as the color of the overcrossing arc, and so that at least two colors are used. The number of colorings meeting these conditions is aknot invariant, independent of the diagram chosen for the link. A trivial link with three components has${\displaystyle n^3-n}$  colorings, obtained from its standard diagram by choosing a color independently for each component and discarding the${\displaystyle n}$  colorings that only use one color. For standard diagram of the Borromean rings, on the other hand, the same pairs of arcs meet at two undercrossings, forcing the arcs that cross over them to have the same color as each other, from which it follows that the only colorings that meet the crossing conditions violate the condition of using more than one color. Because the trivial link has many valid colorings and the Borromean rings have none, they cannot be equivalent.^[4][23]

The Borromean rings are analternating link, as their conventionallink diagram has crossings that alternate between passing over and under each curve, in order along the curve. They are also analgebraic link, a link that can be decomposed byConway spheres into2-tangles. They are the simplest alternating algebraic link which does not have a diagram that is simultaneously alternating and algebraic.^[24] It follows from theTait conjectures that thecrossing number of the Borromean rings (the fewest crossings in any of their link diagrams) is 6, the number of crossings in their alternating diagram.^[4]

The Borromean rings are typically drawn with their rings projecting to circles in the plane of the drawing, but three-dimensional circular Borromean rings are animpossible object: it is not possible to form the Borromean rings from circles in three-dimensional space.^[4] More generallyMichael H. Freedman and Richard Skora (1987) proved using four-dimensionalhyperbolic geometry that no Brunnian link can be exactly circular.^[25] For three rings in their conventional Borromean arrangement, this can be seen from considering thelink diagram. If one assumes that two of the circles 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.^[26] Another argument for the impossibility of circular realizations, byHelge Tverberg, usesinversive geometry to transform any three circles so that one of them becomes a line, making it easier to argue that the other two circles do not link with it to form the Borromean rings.^[27]

However, the Borromean rings can be realized using ellipses.^[2] These may be taken to be ofarbitrarily smalleccentricity: 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. A realization of the Borromean rings by three mutually perpendiculargolden rectangles can be found within a regularicosahedron by connecting three opposite pairs of its edges.^[2] Every three unknottedpolygons in Euclidean space may be combined, after a suitable scaling transformation, to form the Borromean rings. If all three polygons are planar, then scaling is not needed.^[28] In particular, because the Borromean rings can be realized by three triangles, the minimum number of sides possible for each of its loops, thestick number of the Borromean rings is nine.^[29]

More generally,Matthew Cook hasconjectured that any three unknotted simple closed curves in space, not all circles, can be combined without scaling to form the Borromean rings. After Jason Cantarella suggested a possible counterexample, Hugh Nelson Howards weakened the conjecture to apply to any three planar curves that are not all circles. On the other hand, although there are infinitely many Brunnian links with three links, the Borromean rings are the only one that can be formed from three convex curves.^[28]

In knot theory, theropelength of a knot or link is the shortest length of flexible rope (of radius one) that can realize it. Mathematically, such a realization can be described by a smooth curve whose radius-onetubular neighborhood avoids self-intersections. The minimum ropelength of the Borromean rings has not been proven, but the smallest value that has been attained is realized by three copies of a 2-lobed planar curve.^[2][30] Although it resembles an earlier candidate for minimum ropelength, constructed from fourcircular arcs of radius two,^[31] it is slightly modified from that shape, and is composed from 42 smooth pieces defined byelliptic integrals, making it shorter by a fraction of a percent than the piecewise-circular realization. It is this realization, conjectured to minimize ropelength, that was used for theInternational Mathematical Union logo. Its length is${\displaystyle \approx 58.006}$ , while the best proven lower bound on the length is${\displaystyle 12\pi \approx 37.699}$ .^[2][30]

For a discrete analogue of ropelength, the shortest representation using only edges of theinteger lattice, the minimum length for the Borromean rings is exactly${\displaystyle 36}$ . This is the length of a representation using three${\displaystyle 2\times 4}$  integer rectangles, inscribed inJessen's icosahedron in the same way that the representation by golden rectangles is inscribed in the regular icosahedron.^[32]

The Borromean rings are ahyperbolic link: the space surrounding the Borromean rings (theirlink complement) admits a completehyperbolic metric of finite volume. Although hyperbolic links are now considered plentiful, the Borromean rings were one of the earliest examples to be proved hyperbolic, in the 1970s,^[33][34] and this link complement was a central example in the videoNot Knot, produced in 1991 by theGeometry Center.^[35]

Hyperbolic manifolds can be decomposed in a canonical way into gluings of hyperbolic polyhedra (the Epstein–Penner decomposition) and for the Borromean complement this decomposition consists of twoidealregular octahedra.^[34][36] Thevolume of the Borromean complement is${\displaystyle 16\mathrm{\Lambda }(\pi /4)=8G\approx 7.32772\dots }$  where${\displaystyle \mathrm{\Lambda }}$  is theLobachevsky function and${\displaystyle G}$  isCatalan's constant.^[36] The complement of the Borromean rings is universal, in the sense that every closed 3-manifold is abranched cover over this space.^[37]

Inarithmetic topology, there is an analogy betweenknots andprime numbers in which one considers links between primes. The triple of primes(13, 61, 937) are linked modulo 2 (theRédei symbol is −1) but are pairwise unlinked modulo 2 (theLegendre symbols are all 1). Therefore, these primes have been called a "proper Borromean triple modulo 2"^[38] or "mod 2 Borromean primes".^[39]

Amonkey's fist knot is essentially a 3-dimensional representation of the Borromean rings, albeit with three layers, in most cases.^[41] SculptorJohn Robinson has made artworks with threeequilateral triangles made out ofsheet metal, linked to form Borromean rings and resembling a three-dimensional version of the valknut.^[13][29] A common design for a folding wooden tripod consists of three pieces carved from a single piece of wood, with each piece consisting of two lengths of wood, the legs and upper sides of the tripod, connected by two segments of wood that surround an elongated central hole in the piece. Another of the three pieces passes through each of these holes, linking the three pieces together in the Borromean rings pattern. Tripods of this form have been described as coming from Indian or African hand crafts.^[42][43]

In chemistry,molecular Borromean rings are the molecular counterparts of Borromean rings, which aremechanically-interlocked molecular architectures. In 1997, biologist Chengde Mao and coworkers ofNew York University succeeded in constructing a set of rings fromDNA.^[44] In 2003,chemistFraser Stoddart and coworkers atUCLA utilisedcoordination chemistry to construct a set of rings in one step from 18 components.^[40] Borromean ring structures have been used to describe noble metal clusters shielded by a surface layer of thiolate ligands.^[45] A library of Borromean networks has been synthesized by design byGiuseppe Resnati and coworkers viahalogen bond drivenself-assembly.^[46] 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.^[47]

In physics, a quantum-mechanical analog of Borromean rings is called a halo state or anEfimov state, and consists of three bound particles that are not pairwise bound. The existence of such states was predicted by physicistVitaly Efimov in 1970, and confirmed by multiple experiments beginning in 2006.^[48][49] This phenomenon is closely related to aBorromean nucleus, a stable atomic nucleus consisting of three groups of particles that would be unstable in pairs.^[50] Another analog of the Borromean rings inquantum information theory involves the entanglement of threequbits in theGreenberger–Horne–Zeilinger state.^[14]

• Lamb, Evelyn (September 30, 2016),"A few of my favorite spaces: Borromean rings", Roots of Unity,Scientific American
• Borromean Olympic Rings (Brady Haran, 2012),Borromean ribbons (Tadashi Tokieda, 2016), andNeon Knots and Borromean Beer Rings (Clifford Stoll, 2018),Numberphile
• Borromean Rings, International Mathematical Union