1229:
1259:, linked to form Borromean rings and resembling a three-dimensional version of the valknut. 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.
31:
330:
368:
563:
726:
714:
352:
433:
1214:
4095:
253:, 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 equal
986:
4107:
1198:
765:. 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. Another argument for the impossibility of circular realizations, by
527:
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
273:
between above and below when considered in consecutive order around each circle; 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. Two
827:
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.
795:
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. In particular, because the Borromean rings can be realized by three triangles, the minimum number of sides possible for each of its
668:
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
855:
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. Although it resembles an earlier candidate for minimum ropelength, constructed from four
589:, 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.
3229:
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",
1293:
to construct a set of rings in one step from 18 components. Borromean ring structures have been used to describe noble metal clusters shielded by a surface layer of thiolate ligands. A library of
Borromean networks has been synthesized by design by
3142:
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",
1013:
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, and this link complement was a central example in the video
142:
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 a
783:: 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 perpendicular
3258:
Kraemer, T.; Mark, M.; Waldburger, P.; Danzl, J. G.; Chin, C.; Engeser, B.; Lange, A. D.; Pilch, K.; Jaakkola, A.; NĂ€gerl, H.-C.; Grimm, R. (2006), "Evidence for Efimov quantum states in an ultracold gas of caesium atoms",
466:
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").
837:
287:, 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 by
1094:
2641:
Uberti, R.; Janse van
Rensburg, E. J.; Orlandini, E.; Tesi, M. C.; Whittington, S. G. (1998), "Minimal links in the cubic lattice", in Whittington, Stuart G.; Sumners, Witt De; Lodge, Timothy (eds.),
1027:
Hyperbolic manifolds can be decomposed in a canonical way into gluings of hyperbolic polyhedra (the EpsteinâPenner decomposition) and for the Borromean complement this decomposition consists of two
2340:
write that this reference "seems to incorrectly deal only with the case that the three-dimensional configuration has a projection homeomorphic to" the conventional three-circle drawing of the link.
205:. 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. In
851:
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-one
918:
151:
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.
669:
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.
889:
864:, 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 the
828:
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.
773:
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.
971:
1114:
646:
429:; like the Borromean rings these three triangles are linked and not pairwise linked, but this crossing pattern describes a different link than the Borromean rings.
745:
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 an
945:
1138:
666:
448:. 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 by
2890:], Mathematisches Institut, Georg-August-UniversitĂ€t Göttingen: Seminars Winter Term 2004/2005, Göttingen: UniversitĂ€tsdrucke Göttingen, pp. 93â98,
1306:. 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.
1927:
692:. They are the simplest alternating algebraic link which does not have a diagram that is simultaneously alternating and algebraic. It follows from the
516:
502:
2354:
1177:
1267:
761:
that no Brunnian link can be exactly circular. For three rings in their conventional Borromean arrangement, this can be seen from considering the
444:
The Borromean rings have been used in different contexts to indicate strength in unity. In particular, some have used the design to symbolize the
278:) 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.
1713:
1321:, a stable atomic nucleus consisting of three groups of particles that would be unstable in pairs. Another analog of the Borromean rings in
2407:
2142:
1734:
700:
of the Borromean rings (the fewest crossings in any of their link diagrams) is 6, the number of crossings in their alternating diagram.
611:) as the color of the overcrossing arc, and so that at least two colors are used. The number of colorings meeting these conditions is a
421:
in Japan is also decorated with a motif of the Borromean rings, in their conventional circular form. A stone pillar in the 6th-century
3183:(2017), "Halogen bonded Borromean networks by design: topology invariance and metric tuning in a library of multi-component systems",
4153:
1330:
3399:
813:
539:
Some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol in
1041:
171:
3472:
3029:
2731:
2692:
1976:
1563:
1453:
1313:, and consists of three bound particles that are not pairwise bound. The existence of such states was predicted by physicist
1247:
knot is essentially a 3-dimensional representation of the Borromean rings, albeit with three layers, in most cases. Sculptor
997:
formed from two ideal octahedra, seen repeatedly in this view. The rings are infinitely far away, at the octahedron vertices.
462:
was inspired by similar images, although Dante does not detail the geometric arrangement of these circles. The psychoanalyst
4138:
4040:
4143:
3145:
1492:
Cromwell, Peter; Beltrami, Elisabetta; Rampichini, Marta (March 1998), "The Borromean rings", The mathematical tourist,
1463:
4148:
3959:
1184:
are all 1). Therefore, these primes have been called a "proper Borromean triple modulo 2" or "mod 2 Borromean primes".
648:
colorings, obtained from its standard diagram by choosing a color independently for each component and discarding the
1655:
2955:
Chichak, Kelly S.; Cantrill, Stuart J.; Pease, Anthony R.; Chiu, Sheng-Hsien; Cave, Gareth W. V.; Atwood, Jerry L.;
2847:
2569:
2510:
2280:
1433:
865:
841:
680:
has crossings that alternate between passing over and under each curve, in order along the curve. They are also an
512:
3506:
2813:
3325:
2239:
1996:
1580:
1494:
452:. Didron and others have speculated that the description of the Trinity as three equal circles in canto 33 of
3954:
3949:
3825:
2960:
2514:
2311:
2196:
1650:, Mathematics Lecture Series, vol. 7 (2nd ed.), Publish or Perish, Inc., Houston, TX, p. 425,
697:
313:
300:
71:
41:
2675:(1979), "An elliptical path from parabolic representations to hyperbolic structures", in Fenn, Roger (ed.),
1768:
Bruns, Carson J.; Stoddart, J. Fraser (2011), "The mechanical bond: A work of art", in Fabbrizzi, L. (ed.),
1317:, in 1970, and confirmed by multiple experiments beginning in 2006. This phenomenon is closely related to a
4111:
3526:
2672:
1910:
1708:
894:
383:
The name "Borromean rings" comes from the use of these rings, in the form of three linked circles, in the
4163:
4158:
3588:
2573:
2379:
2063:
2037:
860:
of radius two, it is slightly modified from that shape, and is composed from 42 smooth pieces defined by
749:: it is not possible to form the Borromean rings from circles in three-dimensional space. More generally
316:
for the Borromean rings, ".1", is an abbreviated description of the standard link diagram for this link.
3185:
2645:, IMA Volumes in Mathematics and its Applications, vol. 103, New York: Springer, pp. 89â100,
2362:
1322:
780:
977:
in the same way that the representation by golden rectangles is inscribed in the regular icosahedron.
449:
299:, extending earlier listings in the 1920s by Alexander and Briggs, the Borromean rings were given the
3658:
3653:
3594:
3465:
2041:
1263:
1234:
1228:
871:
2764:
2581:
2120:
1248:
1035:
490:
474:, and are still used by the Ballantine brand beer, now distributed by the current brand owner, the
81:
3786:
3344:
2194:
Nanyes, Ollie (October 1993), "An elementary proof that the Borromean rings are non-splittable",
974:
2916:
Morishita, Masanori (2010), "Analogies between knots and primes, 3-manifolds and number rings",
4000:
3969:
3342:
Tanaka, K. (2010), "Observation of a Large Reaction Cross Section in the Drip-Line Nucleus C",
2522:
2462:
Burgiel, H.; Franzblau, D. S.; Gutschera, K. R. (1996), "The mystery of the linked triangles",
2405:
Howards, Hugh Nelson (2013), "Forming the Borromean rings out of arbitrary polygonal unknots",
1994:
Glick, Ned (September 1999), "The 3-ring symbol of Ballantine Beer", The mathematical tourist,
1290:
950:
486:
3441:
3019:
2719:
1645:
1437:
1099:
3830:
2956:
2230:
1964:
1843:, Proceedings of the 2nd International Katachi U Symmetry Symposium, Part 1 (Tsukuba, 1999),
1798:
Lakshminarayan, Arul (May 2007), "Borromean triangles and prime knots in an ancient temple",
1282:
1141:
1010:
618:
475:
422:
374:
288:
190:, both of which have three components bound to each other although no two of them are bound.
162:, but designs based on the Borromean rings have been used in many cultures, including by the
99:
836:
607:
so that at each crossing, the two colors at the undercrossing have the same average (modulo
4133:
4099:
3870:
3458:
3353:
3280:
3179:
Kumar, Vijith; Pilati, Tullio; Terraneo, Giancarlo; Meyer, Franck; Metrangolo, Pierangelo;
3101:
2977:
2939:
2903:
2867:
2741:
2726:, Graduate Texts in Mathematics, vol. 149 (2nd ed.), Springer, pp. 459â461,
2702:
2658:
2620:
2600:
2555:
2493:
2464:
2438:
2375:
2262:
2173:
1896:
1856:
1704:
1690:
1665:
1601:
1545:
1252:
1117:
923:
For a discrete analogue of ropelength, the shortest representation using only edges of the
852:
689:
615:, independent of the diagram chosen for the link. A trivial link with three components has
574:
426:
407:
343:
312:", meaning that this is the second of three 6-crossing 3-component links to be listed. The
258:
3066:
1873:
1732:
Schoeck, Richard J. (Spring 1968), "Mathematics and the languages of literary criticism",
1578:
Chamberland, Marc; Herman, Eugene A. (2015), "Rock-paper-scissors meets Borromean rings",
1541:
8:
3907:
3890:
3404:
2124:
2046:
1800:
1271:
1161:
994:
776:
However, the Borromean rings can be realized using ellipses. These may be taken to be of
758:
545:
507:
329:
234:
135:
30:
3357:
3284:
3105:
2981:
2850:; Montesinos, JosĂ© MarĂa (1983), "The Whitehead link, the Borromean rings and the knot 9
2677:
Topology of Low-Dimensional Manifolds: Proceedings of the Second Sussex Conference, 1977
2604:
1309:
In physics, a quantum-mechanical analog of Borromean rings is called a halo state or an
930:
810:
Are there three unknotted curves, not all circles, that cannot form the Borromean rings?
592:
There are a number of ways of seeing that the Borromean rings are linked. One is to use
3928:
3875:
3489:
3485:
3377:
3304:
3270:
3207:
3125:
3001:
2925:
2789:
2781:
2624:
2590:
2531:
2481:
2442:
2416:
2328:
2213:
2177:
2151:
2099:
2013:
1946:
1817:
1751:
1678:
1605:
1511:
1165:
1144:. The complement of the Borromean rings is universal, in the sense that every closed 3-
1123:
1031:
770:
713:
651:
601:
471:
249:
It is common in mathematics publications that define the Borromean rings to do so as a
139:
2140:
Bai, Sheng; Wang, Weibiao (2020), "New criteria and constructions of Brunnian links",
2038:"Surfaces with edges linked in the same way as the three rings of a well-known design"
1441:
295:
with up to 13 crossings. In the tables of knots and links in Dale Rolfsen's 1976 book
261:, close enough together that their interiors have a common intersection (such as in a
4025:
3974:
3924:
3880:
3840:
3835:
3753:
3369:
3296:
3212:
3162:
3117:
3045:
3025:
2993:
2968:
2727:
2688:
2446:
2309:
Lindström, Bernt; Zetterström, Hans-Olov (1991), "Borromean circles are impossible",
2181:
2017:
1972:
1821:
1781:
1681:(1970), "An enumeration of knots and links, and some of their algebraic properties",
1651:
1559:
1515:
1449:
1318:
861:
777:
746:
533:
425:
in India shows three equilateral triangles rotated from each other to form a regular
391:
266:
187:
155:
51:
3005:
2793:
485:
to include the Borromean rings was a catalog of knots and links compiled in 1876 by
367:
4060:
3885:
3781:
3516:
3381:
3365:
3361:
3308:
3288:
3261:
3239:
3202:
3194:
3180:
3154:
3129:
3109:
3092:
2985:
2891:
2809:
2773:
2680:
2646:
2608:
2541:
2477:
2473:
2426:
2320:
2289:
2275:
2248:
2205:
2161:
2091:
2005:
1884:
1809:
1773:
1743:
1589:
1551:
1503:
1352:
1295:
1244:
1204:
1028:
784:
750:
732:
693:
673:
562:
270:
222:
202:
148:
114:
2628:
2234:
1840:
1609:
274:
links are said to be equivalent if there is a continuous deformation of space (an
4020:
3984:
3919:
3865:
3820:
3813:
3703:
3498:
3423:
3321:
2935:
2899:
2863:
2737:
2698:
2654:
2616:
2551:
2489:
2434:
2371:
2258:
2169:
1892:
1852:
1686:
1661:
1597:
1555:
1467:
1181:
1021:
1006:
1002:
990:
924:
529:
498:
453:
437:
275:
230:
118:
3450:
3090:
Mao, C.; Sun, W.; Seeman, N. C. (1997), "Assembly of Borromean rings from DNA",
2650:
1888:
725:
418:
4080:
3979:
3941:
3860:
3773:
3648:
3640:
3600:
3431:
2823:
2350:
2033:
1630:
1149:
766:
681:
612:
593:
566:
494:
395:
283:
226:
210:
3427:
3419:
3411:
2881:
2612:
2430:
2165:
1813:
1593:
4127:
4015:
3803:
3796:
3791:
3166:
3158:
2895:
2294:
2253:
1694:; see description of notation, pp. 332â333, and second line of table, p. 348.
1537:
1460:
1314:
1303:
1220:
685:
586:
570:
540:
463:
457:
218:
2989:
2546:
432:
4030:
4010:
3914:
3897:
3693:
3630:
3373:
3300:
3216:
2997:
1785:
1310:
1299:
1169:
857:
820:
797:
762:
677:
384:
262:
250:
238:
183:
159:
144:
61:
3121:
2679:, Lecture Notes in Mathematics, vol. 722, Springer, pp. 99â133,
1213:
478:. For this reason they have sometimes been called the "Ballantine rings".
166:
and in Japan. They have been used in Christian symbolism as a sign of the
4045:
3808:
3713:
3582:
3562:
3552:
3544:
3536:
3481:
3435:
3415:
3275:
1883:, Boston Studies in the Philosophy of Science, Springer, pp. 53â59,
1777:
1256:
788:
736:
582:
482:
414:
388:
351:
206:
198:
127:
3292:
2103:
1950:
1772:, Topics in Current Chemistry, vol. 323, Springer, pp. 19â72,
569:
diagram for the Borromean rings. The vertical dotted black midline is a
4065:
4050:
4005:
3902:
3855:
3850:
3845:
3675:
3572:
3244:
3198:
2785:
2684:
2485:
2332:
2217:
2009:
1507:
848:
824:
802:
791:
by connecting three opposite pairs of its edges. Every three unknotted
292:
178:. Physical instances of the Borromean rings have been made from linked
2817:
1755:
1683:
Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967)
4070:
3738:
3113:
2640:
2595:
2536:
2095:
2777:
2324:
2209:
398:. The link itself is much older and has appeared in the form of the
4055:
3665:
3073:, Centre for the Popularisation of Maths, University of Wales, 2002
2156:
1747:
1197:
1145:
1016:
985:
411:
209:, the Borromean rings can be proved to be linked by counting their
163:
158:, who used the circular form of these rings as an element of their
2930:
2421:
2061:
Gardner, Martin (September 1978), "The Toroids of Dr. Klonefake",
1446:
Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture
1172:
in which one considers links between primes. The triple of primes
2278:; Skora, Richard (1987), "Strange actions of groups on spheres",
1279:
792:
519:
in Madrid, Spain to use a new logo based on the Borromean rings.
445:
401:
358:
337:
194:
167:
4075:
3723:
3683:
254:
2108:
the stone bears also representations of three horns interlaced
1373:
3964:
3228:
3141:
1326:
1266:
are the molecular counterparts of Borromean rings, which are
193:
Geometrically, the Borromean rings may be realized by linked
600:, colorings of the arcs of a link diagram with the integers
4035:
1962:
1376:
1364:
1286:
3257:
2508:
1625:
1491:
1358:
1089:{\displaystyle 16\Lambda (\pi /4)=8G\approx 7.32772\dots }
241:
have analogous linking properties to the Borromean rings.
2888:
Massey products in the Galois cohomology of number fields
2461:
1275:
179:
3178:
2954:
2567:
2055:
The Unexpected Hanging and Other Mathematical Diversions
927:, the minimum length for the Borromean rings is exactly
2883:
Masseyprodukte in der Galoiskohomologie von Zahlkörpern
1915:(in French), Paris: Imprimerie Royale, pp. 568â569
1544:(2018), "Chapter 15: The Borromean Rings Don't Exist",
3326:"Strange physical theory proved after nearly 40 years"
2845:
3046:"Gathering clues from Margot's extraordinary objects"
2053:
Gardner, Martin (1991), "Knots and Borromean Rings",
1963:
Ragland-Sullivan, Ellie; Milovanovic, Dragan (2004),
1438:"The Borromean Rings: A video about the New IMU logo"
1379:
1126:
1102:
1044:
953:
947:. This is the length of a representation using three
933:
897:
891:, while the best proven lower bound on the length is
874:
654:
621:
2308:
1881:
Potentiality, Entanglement and Passion-at-a-Distance
1361:
1355:
1370:
1367:
1005:: the space surrounding the Borromean rings (their
182:or other molecules, and they have analogues in the
2509:Cantarella, Jason; Fu, Joseph H. G.; Kusner, Rob;
1879:, in Cohen, R. S.; Horne, M.; Stachel, J. (eds.),
1270:. In 1997, biologist Chengde Mao and coworkers of
1132:
1108:
1088:
965:
939:
912:
883:
660:
640:
3480:
1577:
1219:Borromean ring knitting project by knot theorist
4125:
2808:
1448:, London: Tarquin Publications, pp. 63â70,
1268:mechanically-interlocked molecular architectures
585:, the Borromean rings are a simple example of a
154:The Borromean rings are named after the Italian
149:alternatingly crossing over and under each other
1180:is â1) but are pairwise unlinked modulo 2 (the
804:
501:for the Borromean rings in his September 1961 "
399:
335:
3400:"A few of my favorite spaces: Borromean rings"
2574:"On the minimum ropelength of knots and links"
2504:
2502:
2235:"On the algebraic part of an alternating link"
2229:
1797:
1274:succeeded in constructing a set of rings from
3466:
2274:
2057:, University of Chicago Press, pp. 24â33
1925:
1767:
1714:Proceedings of the Royal Society of Edinburgh
1536:
754:
719:Realization of Borromean rings using ellipses
522:
517:25th International Congress of Mathematicians
3089:
2819:The Geometry and Topology of Three-Manifolds
2713:
2711:
2457:
2455:
2408:Journal of Knot Theory and Its Ramifications
2400:
2398:
2337:
2143:Journal of Knot Theory and Its Ramifications
1431:
2950:
2948:
2499:
1735:The Journal of Aesthetics and Art Criticism
1550:(6th ed.), Springer, pp. 99â106,
1461:The Borromean Rings: A new logo for the IMU
3473:
3459:
2515:"Criticality for the Gehring link problem"
2302:
2028:
2026:
1703:
552:
493:, the Borromean rings were popularized by
244:
29:
3320:
3274:
3243:
3206:
2929:
2915:
2909:
2839:
2804:
2802:
2717:
2708:
2594:
2561:
2545:
2535:
2452:
2420:
2395:
2343:
2293:
2252:
2223:
2155:
1874:"Borromean entanglement of the GHZ state"
1791:
1697:
1487:
1485:
1483:
1481:
1479:
16:Three linked but pairwise separated rings
3314:
3135:
2945:
2643:Topology and Geometry in Polymer Science
2349:
2133:
1989:
1987:
1902:
1761:
1408:The Pronunciation of 10,000 Proper Names
1187:
984:
835:
561:
431:
361:, adapted from a 13th-century manuscript
319:
265:or the three circles used to define the
170:, and in modern commerce as the logo of
3335:
3222:
3172:
3059:
3043:
3037:
2747:
2634:
2404:
2268:
2139:
2113:
2071:
2060:
2052:
2032:
2023:
1871:
1834:
1832:
1830:
1731:
1725:
1643:
1620:
1618:
1571:
814:(more unsolved problems in mathematics)
4126:
3341:
3251:
3050:Tewkesbury Historical Society Bulletin
3017:
3011:
2873:
2799:
2754:Abbott, Steve (July 1997), "Review of
2753:
2568:Cantarella, Jason; Kusner, Robert B.;
2193:
2187:
2129:(4th ed.), March 1970, p. 43
2078:Baird, Joseph L. (1970), "Unferth the
1965:"Introduction: Topologically Speaking"
1926:Saiber, Arielle; Mbirika, aBa (2013),
1908:
1867:
1865:
1838:
1685:, Oxford: Pergamon, pp. 329â358,
1677:
1637:
1532:
1530:
1528:
1526:
1524:
1476:
980:
197:, or (using the vertices of a regular
3454:
2879:
2671:
2665:
2077:
2067:, vol. 2, no. 5, p. 29
1993:
1984:
1956:
1919:
1671:
1427:
1425:
1423:
1421:
1419:
1417:
1415:
4106:
3397:
2299:; see in particular Lemma 3.2, p. 89
1827:
1615:
913:{\displaystyle 12\pi \approx 37.699}
417:dating back to the 7th century. The
138:in three-dimensional space that are
3428:Neon Knots and Borromean Beer Rings
3398:Lamb, Evelyn (September 30, 2016),
3146:The Journal of Physical Chemistry C
3083:
2724:Foundations of Hyperbolic Manifolds
1862:
1709:"On a case of interlacing surfaces"
1521:
1325:involves the entanglement of three
684:, a link that can be decomposed by
470:The rings were used as the logo of
174:, giving them the alternative name
13:
3446:, International Mathematical Union
1412:
1103:
1048:
14:
4175:
3391:
3018:Ashley, Clifford Warren (1993) ,
2856:Seminario MatemĂĄtico de Barcelona
1909:Didron, Adolphe Napoléon (1843),
1331:GreenbergerâHorneâZeilinger state
973:integer rectangles, inscribed in
753: and Richard Skora (
4154:Non-tricolorable knots and links
4105:
4094:
4093:
2720:"The Borromean rings complement"
2281:Journal of Differential Geometry
1472:International Mathematical Union
1351:
1227:
1212:
1196:
1155:
866:International Mathematical Union
842:International Mathematical Union
800:of the Borromean rings is nine.
757:) proved using four-dimensional
724:
712:
513:International Mathematical Union
366:
350:
328:
1038:of the Borromean complement is
805:Unsolved problem in mathematics
3960:DowkerâThistlethwaite notation
3366:10.1103/PhysRevLett.104.062701
2478:10.1080/0025570x.1996.11996399
2240:Pacific Journal of Mathematics
1997:The Mathematical Intelligencer
1841:"Are Borromean links so rare?"
1581:The Mathematical Intelligencer
1495:The Mathematical Intelligencer
1400:
1343:
1164:, there is an analogy between
1065:
1051:
884:{\displaystyle \approx 58.006}
787:can be found within a regular
543:, based on a depiction in the
1:
2822:, p. 165, archived from
2312:American Mathematical Monthly
2197:American Mathematical Monthly
1969:Lacan: Topologically Speaking
1393:
1251:has made artworks with three
831:
703:
557:
257:centered at the points of an
2513:; Wrinkle, Nancy C. (2006),
1556:10.1007/978-3-662-57265-8_15
573:separating the diagram into
7:
4139:Alternating knots and links
3232:Organic Chemistry Frontiers
2961:"Molecular Borromean rings"
2718:Ratcliffe, John G. (2006),
2651:10.1007/978-1-4612-1712-1_9
1889:10.1007/978-94-017-2732-7_4
1459:; see the video itself at "
672:The Borromean rings are an
10:
4180:
4144:Hyperbolic knots and links
3024:, Doubleday, p. 354,
2814:"7. Computation of volume"
2363:The Mathematical Scientist
2338:Gunn & Sullivan (2008)
1839:Jablan, Slavik V. (1999),
1323:quantum information theory
1020:, produced in 1991 by the
1001:The Borromean rings are a
993:of the Borromean rings, a
523:Partial and multiple rings
4149:Unfibered knots and links
4089:
3993:
3950:AlexanderâBriggs notation
3937:
3772:
3674:
3639:
3497:
3067:"African Borromean Rings"
2613:10.1007/s00222-002-0234-y
2431:10.1142/S0218216513500831
2231:Thistlethwaite, Morwen B.
2166:10.1142/S0218216520430087
1814:10.1007/s12045-007-0049-7
1594:10.1007/s00283-014-9499-4
1406:Mackey & Mackay 1922
1264:molecular Borromean rings
1235:Molecular Borromean rings
1176:are linked modulo 2 (the
966:{\displaystyle 2\times 4}
301:AlexanderâBriggs notation
113:
108:
98:
80:
70:
60:
50:
40:
28:
23:
3159:10.1021/acs.jpcc.5b08193
3021:The Ashley Book of Knots
2896:10.11588/heidok.00004418
2765:The Mathematical Gazette
2582:Inventiones Mathematicae
2254:10.2140/pjm.1991.151.317
2064:Asimov's Science Fiction
1336:
1109:{\displaystyle \Lambda }
676:, as their conventional
491:recreational mathematics
410:with parallel sides, on
373:Linked triangles in the
357:Symbol of the Christian
4041:List of knots and links
3589:KinoshitaâTerasaka knot
3412:Borromean Olympic Rings
3345:Physical Review Letters
2990:10.1126/science.1096914
2547:10.2140/gt.2006.10.2055
2523:Geometry & Topology
1912:Iconographie Chrétienne
1872:Aravind, P. K. (1997),
751:Michael H. Freedman
641:{\displaystyle n^{3}-n}
553:Mathematical properties
450:Adolphe Napoléon Didron
400:
336:
245:Definition and notation
2760:Supplement to Not Knot
2295:10.4310/jdg/1214440725
1644:Rolfsen, Dale (1990),
1291:coordination chemistry
1134:
1110:
1090:
998:
967:
941:
914:
885:
844:
662:
642:
578:
441:
440:of the Borromean rings
3831:Finite type invariant
3324:(December 16, 2009),
3071:Mathematics and Knots
3044:Freeman, Jim (2015),
2880:Vogel, Denis (2005),
1705:Crum Brown, Alexander
1440:, in Sarhangi, Reza;
1253:equilateral triangles
1188:Physical realizations
1135:
1111:
1091:
988:
968:
942:
915:
886:
839:
663:
643:
565:
476:Pabst Brewing Company
435:
423:Marundeeswarar Temple
408:equilateral triangles
375:Marundeeswarar Temple
344:Stora Hammars I stone
320:History and symbolism
289:Morwen Thistlethwaite
237:, certain triples of
217:. As links, they are
2848:Lozano, MarĂa Teresa
2662:; see Table 2, p. 97
2465:Mathematics Magazine
2355:"On Borromean rings"
2336:. Note however that
2276:Freedman, Michael H.
1778:10.1007/128_2011_296
1547:Proofs from THE BOOK
1124:
1118:Lobachevsky function
1100:
1042:
1009:) admits a complete
975:Jessen's icosahedron
951:
931:
895:
872:
868:logo. Its length is
853:tubular neighborhood
847:In knot theory, the
652:
619:
259:equilateral triangle
140:topologically linked
136:simple closed curves
4001:Alexander's theorem
3405:Scientific American
3358:2010PhRvL.104f2701T
3293:10.1038/nature04626
3285:2006Natur.440..315K
3153:(49): 27768â27785,
3106:1997Natur.386..137M
2982:2004Sci...304.1308C
2976:(5675): 1308â1312,
2957:Stoddart, J. Fraser
2605:2002InMat.150..257C
2415:(14): 1350083, 15,
2150:(13): 2043008, 27,
2126:Principia Discordia
2047:Scientific American
1770:Beauty in Chemistry
1272:New York University
1162:arithmetic topology
995:hyperbolic manifold
981:Hyperbolic geometry
759:hyperbolic geometry
546:Principia Discordia
508:Scientific American
235:arithmetic topology
4164:Impossible objects
4159:Geometric topology
3402:, Roots of Unity,
3245:10.1039/c6qo00025h
3199:10.1039/C6SC04478F
2918:Sugaku Expositions
2685:10.1007/BFb0063194
2042:Mathematical Games
2036:(September 1961),
2010:10.1007/bf03025332
1542:Ziegler, GĂŒnter M.
1508:10.1007/bf03024401
1466:2021-03-08 at the
1298:and coworkers via
1142:Catalan's constant
1130:
1106:
1086:
999:
963:
940:{\displaystyle 36}
937:
910:
881:
862:elliptic integrals
845:
771:inversive geometry
658:
638:
579:
503:Mathematical Games
481:The first work of
442:
4121:
4120:
3975:Reidemeister move
3841:Khovanov homology
3836:Hyperbolic volume
3420:Borromean ribbons
3269:(7082): 315â318,
3181:Resnati, Giuseppe
3100:(6621): 137â138,
3031:978-0-385-04025-9
2846:Hilden, Hugh M.;
2733:978-0-387-33197-3
2694:978-3-540-09506-4
2570:Sullivan, John M.
2511:Sullivan, John M.
1978:978-1-892746-76-4
1707:(December 1885),
1565:978-3-662-57265-8
1455:978-0-9665201-9-4
1434:Sullivan, John M.
1319:Borromean nucleus
1285:and coworkers at
1152:over this space.
1133:{\displaystyle G}
1032:regular octahedra
785:golden rectangles
778:arbitrarily small
747:impossible object
733:golden rectangles
661:{\displaystyle n}
534:Diana of Poitiers
291:in a list of all
269:). Its crossings
267:Reuleaux triangle
203:golden rectangles
156:House of Borromeo
124:
123:
52:Hyperbolic volume
4171:
4109:
4108:
4097:
4096:
4061:Tait conjectures
3764:
3763:
3749:
3748:
3734:
3733:
3626:
3625:
3611:
3610:
3595:(â2,3,7) pretzel
3475:
3468:
3461:
3452:
3451:
3447:
3408:
3385:
3384:
3339:
3333:
3332:
3322:Moskowitz, Clara
3318:
3312:
3311:
3278:
3276:cond-mat/0512394
3255:
3249:
3248:
3247:
3226:
3220:
3219:
3210:
3193:(3): 1801â1810,
3186:Chemical Science
3176:
3170:
3169:
3139:
3133:
3132:
3114:10.1038/386137b0
3087:
3081:
3080:
3079:
3078:
3063:
3057:
3056:
3041:
3035:
3034:
3015:
3009:
3008:
2965:
2959:(May 28, 2004),
2952:
2943:
2942:
2933:
2913:
2907:
2906:
2877:
2871:
2870:
2854:are universal",
2843:
2837:
2836:
2835:
2834:
2828:
2810:William Thurston
2806:
2797:
2796:
2772:(491): 340â342,
2751:
2745:
2744:
2715:
2706:
2705:
2669:
2663:
2661:
2638:
2632:
2631:
2598:
2578:
2565:
2559:
2558:
2549:
2539:
2530:(4): 2055â2116,
2519:
2506:
2497:
2496:
2459:
2450:
2449:
2424:
2402:
2393:
2392:
2391:
2390:
2384:
2378:, archived from
2359:
2347:
2341:
2335:
2306:
2300:
2298:
2297:
2272:
2266:
2265:
2256:
2227:
2221:
2220:
2191:
2185:
2184:
2159:
2137:
2131:
2130:
2117:
2111:
2110:
2096:10.2307/43631234
2075:
2069:
2068:
2058:
2050:
2030:
2021:
2020:
1991:
1982:
1981:
1960:
1954:
1953:
1945:(131): 237â272,
1940:
1923:
1917:
1916:
1906:
1900:
1899:
1878:
1869:
1860:
1859:
1836:
1825:
1824:
1795:
1789:
1788:
1765:
1759:
1758:
1729:
1723:
1722:
1701:
1695:
1693:
1675:
1669:
1668:
1641:
1635:
1622:
1613:
1612:
1575:
1569:
1568:
1534:
1519:
1518:
1489:
1474:
1458:
1442:SĂ©quin, Carlo H.
1429:
1410:
1404:
1387:
1386:
1385:
1382:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1347:
1296:Giuseppe Resnati
1231:
1216:
1200:
1182:Legendre symbols
1175:
1139:
1137:
1136:
1131:
1115:
1113:
1112:
1107:
1095:
1093:
1092:
1087:
1061:
972:
970:
969:
964:
946:
944:
943:
938:
919:
917:
916:
911:
890:
888:
887:
882:
819:More generally,
806:
728:
716:
694:Tait conjectures
674:alternating link
667:
665:
664:
659:
647:
645:
644:
639:
631:
630:
610:
605:
597:
499:Seifert surfaces
405:
370:
354:
341:
332:
311:
310:
214:
188:Borromean nuclei
176:Ballantine rings
94:
93:
33:
21:
20:
4179:
4178:
4174:
4173:
4172:
4170:
4169:
4168:
4124:
4123:
4122:
4117:
4085:
3989:
3955:Conway notation
3939:
3933:
3920:Tricolorability
3768:
3762:
3759:
3758:
3757:
3747:
3744:
3743:
3742:
3732:
3729:
3728:
3727:
3719:
3709:
3699:
3689:
3670:
3649:Composite knots
3635:
3624:
3621:
3620:
3619:
3616:Borromean rings
3609:
3606:
3605:
3604:
3578:
3568:
3558:
3548:
3540:
3532:
3522:
3512:
3493:
3479:
3443:Borromean Rings
3440:
3424:Tadashi Tokieda
3394:
3389:
3388:
3340:
3336:
3319:
3315:
3256:
3252:
3227:
3223:
3177:
3173:
3140:
3136:
3088:
3084:
3076:
3074:
3065:
3064:
3060:
3042:
3038:
3032:
3016:
3012:
2963:
2953:
2946:
2914:
2910:
2878:
2874:
2853:
2844:
2840:
2832:
2830:
2826:
2807:
2800:
2778:10.2307/3619248
2752:
2748:
2734:
2716:
2709:
2695:
2670:
2666:
2639:
2635:
2576:
2566:
2562:
2517:
2507:
2500:
2460:
2453:
2403:
2396:
2388:
2386:
2382:
2357:
2351:Tverberg, Helge
2348:
2344:
2325:10.2307/2323803
2307:
2303:
2273:
2269:
2228:
2224:
2210:10.2307/2324788
2192:
2188:
2138:
2134:
2119:
2118:
2114:
2076:
2072:
2051:, reprinted as
2034:Gardner, Martin
2031:
2024:
1992:
1985:
1979:
1971:, Other Press,
1961:
1957:
1938:
1924:
1920:
1907:
1903:
1876:
1870:
1863:
1837:
1828:
1796:
1792:
1766:
1762:
1730:
1726:
1702:
1698:
1676:
1672:
1658:
1647:Knots and Links
1642:
1638:
1626:Borromean rings
1623:
1616:
1576:
1572:
1566:
1535:
1522:
1490:
1477:
1468:Wayback Machine
1456:
1432:Gunn, Charles;
1430:
1413:
1405:
1401:
1396:
1391:
1390:
1354:
1350:
1348:
1344:
1339:
1283:Fraser Stoddart
1241:
1240:
1239:
1238:
1237:
1232:
1224:
1223:
1217:
1209:
1208:
1201:
1190:
1173:
1158:
1125:
1122:
1121:
1101:
1098:
1097:
1057:
1043:
1040:
1039:
1022:Geometry Center
1007:link complement
1003:hyperbolic link
983:
952:
949:
948:
932:
929:
928:
925:integer lattice
896:
893:
892:
873:
870:
869:
834:
817:
816:
811:
808:
743:
742:
741:
740:
739:
729:
721:
720:
717:
706:
698:crossing number
653:
650:
649:
626:
622:
620:
617:
616:
608:
603:
595:
560:
555:
530:Snoldelev stone
525:
515:decided at the
511:. In 2006, the
497:, who featured
472:Ballantine beer
438:Seifert surface
406:, three linked
381:
380:
379:
378:
377:
371:
363:
362:
355:
347:
346:
333:
322:
314:Conway notation
309:
306:
305:
304:
297:Knots and Links
276:ambient isotopy
247:
212:
172:Ballantine beer
132:Borromean rings
92:
89:
88:
87:
72:Conway notation
36:
24:Borromean rings
17:
12:
11:
5:
4177:
4167:
4166:
4161:
4156:
4151:
4146:
4141:
4136:
4119:
4118:
4116:
4115:
4103:
4090:
4087:
4086:
4084:
4083:
4081:Surgery theory
4078:
4073:
4068:
4063:
4058:
4053:
4048:
4043:
4038:
4033:
4028:
4023:
4018:
4013:
4008:
4003:
3997:
3995:
3991:
3990:
3988:
3987:
3982:
3980:Skein relation
3977:
3972:
3967:
3962:
3957:
3952:
3946:
3944:
3935:
3934:
3932:
3931:
3925:Unknotting no.
3922:
3917:
3912:
3911:
3910:
3900:
3895:
3894:
3893:
3888:
3883:
3878:
3873:
3863:
3858:
3853:
3848:
3843:
3838:
3833:
3828:
3823:
3818:
3817:
3816:
3806:
3801:
3800:
3799:
3789:
3784:
3778:
3776:
3770:
3769:
3767:
3766:
3760:
3751:
3745:
3736:
3730:
3721:
3717:
3711:
3707:
3701:
3697:
3691:
3687:
3680:
3678:
3672:
3671:
3669:
3668:
3663:
3662:
3661:
3656:
3645:
3643:
3637:
3636:
3634:
3633:
3628:
3622:
3613:
3607:
3598:
3592:
3586:
3580:
3576:
3570:
3566:
3560:
3556:
3550:
3546:
3542:
3538:
3534:
3530:
3524:
3520:
3514:
3510:
3503:
3501:
3495:
3494:
3478:
3477:
3470:
3463:
3455:
3449:
3448:
3438:
3432:Clifford Stoll
3409:
3393:
3392:External links
3390:
3387:
3386:
3334:
3313:
3250:
3238:(6): 667â672,
3221:
3171:
3134:
3082:
3058:
3036:
3030:
3010:
2944:
2908:
2872:
2851:
2838:
2812:(March 2002),
2798:
2746:
2732:
2707:
2693:
2664:
2633:
2589:(2): 257â286,
2560:
2498:
2451:
2394:
2342:
2319:(4): 340â341,
2301:
2267:
2247:(2): 317â333,
2222:
2204:(8): 786â789,
2186:
2132:
2112:
2070:
2022:
1983:
1977:
1955:
1918:
1901:
1861:
1851:(4): 269â277,
1826:
1790:
1760:
1748:10.2307/429121
1742:(3): 367â376,
1724:
1696:
1670:
1656:
1636:
1631:The Knot Atlas
1614:
1570:
1564:
1538:Aigner, Martin
1520:
1475:
1454:
1411:
1398:
1397:
1395:
1392:
1389:
1388:
1341:
1340:
1338:
1335:
1262:In chemistry,
1233:
1226:
1225:
1218:
1211:
1210:
1202:
1195:
1194:
1193:
1192:
1191:
1189:
1186:
1157:
1154:
1150:branched cover
1129:
1105:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1060:
1056:
1053:
1050:
1047:
982:
979:
962:
959:
956:
936:
909:
906:
903:
900:
880:
877:
833:
830:
812:
809:
803:
767:Helge Tverberg
730:
723:
722:
718:
711:
710:
709:
708:
707:
705:
702:
686:Conway spheres
682:algebraic link
657:
637:
634:
629:
625:
613:knot invariant
567:Algebraic link
559:
556:
554:
551:
532:horns and the
524:
521:
495:Martin Gardner
396:Northern Italy
372:
365:
364:
356:
349:
348:
334:
327:
326:
325:
324:
323:
321:
318:
307:
284:The Knot Atlas
246:
243:
122:
121:
111:
110:
106:
105:
102:
100:Thistlethwaite
96:
95:
90:
84:
78:
77:
74:
68:
67:
64:
58:
57:
54:
48:
47:
44:
38:
37:
34:
26:
25:
15:
9:
6:
4:
3:
2:
4176:
4165:
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4131:
4129:
4114:
4113:
4104:
4102:
4101:
4092:
4091:
4088:
4082:
4079:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4032:
4029:
4027:
4024:
4022:
4019:
4017:
4016:Conway sphere
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3998:
3996:
3992:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3951:
3948:
3947:
3945:
3943:
3936:
3930:
3926:
3923:
3921:
3918:
3916:
3913:
3909:
3906:
3905:
3904:
3901:
3899:
3896:
3892:
3889:
3887:
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3868:
3867:
3864:
3862:
3859:
3857:
3854:
3852:
3849:
3847:
3844:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3822:
3819:
3815:
3812:
3811:
3810:
3807:
3805:
3802:
3798:
3795:
3794:
3793:
3790:
3788:
3787:Arf invariant
3785:
3783:
3780:
3779:
3777:
3775:
3771:
3755:
3752:
3740:
3737:
3725:
3722:
3715:
3712:
3705:
3702:
3695:
3692:
3685:
3682:
3681:
3679:
3677:
3673:
3667:
3664:
3660:
3657:
3655:
3652:
3651:
3650:
3647:
3646:
3644:
3642:
3638:
3632:
3629:
3617:
3614:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3574:
3571:
3564:
3561:
3554:
3551:
3549:
3543:
3541:
3535:
3528:
3525:
3518:
3515:
3508:
3505:
3504:
3502:
3500:
3496:
3491:
3487:
3483:
3476:
3471:
3469:
3464:
3462:
3457:
3456:
3453:
3445:
3444:
3439:
3437:
3433:
3429:
3426:, 2016), and
3425:
3421:
3417:
3413:
3410:
3407:
3406:
3401:
3396:
3395:
3383:
3379:
3375:
3371:
3367:
3363:
3359:
3355:
3352:(6): 062701,
3351:
3347:
3346:
3338:
3331:
3327:
3323:
3317:
3310:
3306:
3302:
3298:
3294:
3290:
3286:
3282:
3277:
3272:
3268:
3264:
3263:
3254:
3246:
3241:
3237:
3233:
3225:
3218:
3214:
3209:
3204:
3200:
3196:
3192:
3188:
3187:
3182:
3175:
3168:
3164:
3160:
3156:
3152:
3148:
3147:
3138:
3131:
3127:
3123:
3119:
3115:
3111:
3107:
3103:
3099:
3095:
3094:
3086:
3072:
3068:
3062:
3055:
3051:
3047:
3040:
3033:
3027:
3023:
3022:
3014:
3007:
3003:
2999:
2995:
2991:
2987:
2983:
2979:
2975:
2971:
2970:
2962:
2958:
2951:
2949:
2941:
2937:
2932:
2927:
2923:
2919:
2912:
2905:
2901:
2897:
2893:
2889:
2885:
2884:
2876:
2869:
2865:
2861:
2857:
2849:
2842:
2829:on 2020-07-27
2825:
2821:
2820:
2815:
2811:
2805:
2803:
2795:
2791:
2787:
2783:
2779:
2775:
2771:
2767:
2766:
2761:
2757:
2750:
2743:
2739:
2735:
2729:
2725:
2721:
2714:
2712:
2704:
2700:
2696:
2690:
2686:
2682:
2678:
2674:
2673:Riley, Robert
2668:
2660:
2656:
2652:
2648:
2644:
2637:
2630:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2597:
2592:
2588:
2584:
2583:
2575:
2571:
2564:
2557:
2553:
2548:
2543:
2538:
2533:
2529:
2525:
2524:
2516:
2512:
2505:
2503:
2495:
2491:
2487:
2483:
2479:
2475:
2472:(2): 94â102,
2471:
2467:
2466:
2458:
2456:
2448:
2444:
2440:
2436:
2432:
2428:
2423:
2418:
2414:
2410:
2409:
2401:
2399:
2385:on 2021-03-16
2381:
2377:
2373:
2369:
2365:
2364:
2356:
2352:
2346:
2339:
2334:
2330:
2326:
2322:
2318:
2314:
2313:
2305:
2296:
2291:
2287:
2283:
2282:
2277:
2271:
2264:
2260:
2255:
2250:
2246:
2242:
2241:
2236:
2232:
2226:
2219:
2215:
2211:
2207:
2203:
2199:
2198:
2190:
2183:
2179:
2175:
2171:
2167:
2163:
2158:
2153:
2149:
2145:
2144:
2136:
2128:
2127:
2122:
2116:
2109:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2074:
2066:
2065:
2056:
2049:
2048:
2043:
2039:
2035:
2029:
2027:
2019:
2015:
2011:
2007:
2003:
1999:
1998:
1990:
1988:
1980:
1974:
1970:
1966:
1959:
1952:
1948:
1944:
1943:Dante Studies
1937:
1935:
1931:
1922:
1914:
1913:
1905:
1898:
1894:
1890:
1886:
1882:
1875:
1868:
1866:
1858:
1854:
1850:
1846:
1842:
1835:
1833:
1831:
1823:
1819:
1815:
1811:
1807:
1803:
1802:
1794:
1787:
1783:
1779:
1775:
1771:
1764:
1757:
1753:
1749:
1745:
1741:
1737:
1736:
1728:
1720:
1716:
1715:
1710:
1706:
1700:
1692:
1688:
1684:
1680:
1679:Conway, J. H.
1674:
1667:
1663:
1659:
1657:0-914098-16-0
1653:
1649:
1648:
1640:
1633:
1632:
1627:
1621:
1619:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1582:
1574:
1567:
1561:
1557:
1553:
1549:
1548:
1543:
1539:
1533:
1531:
1529:
1527:
1525:
1517:
1513:
1509:
1505:
1501:
1497:
1496:
1488:
1486:
1484:
1482:
1480:
1473:
1469:
1465:
1462:
1457:
1451:
1447:
1443:
1439:
1435:
1428:
1426:
1424:
1422:
1420:
1418:
1416:
1409:
1403:
1399:
1384:
1346:
1342:
1334:
1332:
1328:
1324:
1320:
1316:
1315:Vitaly Efimov
1312:
1307:
1305:
1304:self-assembly
1301:
1297:
1292:
1288:
1284:
1281:
1277:
1273:
1269:
1265:
1260:
1258:
1254:
1250:
1249:John Robinson
1246:
1245:monkey's fist
1236:
1230:
1222:
1221:Laura Taalman
1215:
1206:
1205:monkey's fist
1199:
1185:
1183:
1179:
1174:(13, 61, 937)
1171:
1170:prime numbers
1167:
1163:
1156:Number theory
1153:
1151:
1147:
1143:
1127:
1119:
1083:
1080:
1077:
1074:
1071:
1068:
1062:
1058:
1054:
1045:
1037:
1033:
1030:
1025:
1023:
1019:
1018:
1012:
1008:
1004:
996:
992:
987:
978:
976:
960:
957:
954:
934:
926:
921:
907:
904:
901:
898:
878:
875:
867:
863:
859:
858:circular arcs
854:
850:
843:
838:
829:
826:
822:
815:
801:
799:
794:
790:
786:
782:
779:
774:
772:
768:
764:
760:
756:
752:
748:
738:
735:in a regular
734:
731:Three linked
727:
715:
701:
699:
695:
691:
687:
683:
679:
675:
670:
655:
635:
632:
627:
623:
614:
606:
599:
590:
588:
587:Brunnian link
584:
576:
572:
571:Conway sphere
568:
564:
550:
548:
547:
542:
541:Discordianism
537:
535:
531:
520:
518:
514:
510:
509:
504:
500:
496:
492:
488:
484:
479:
477:
473:
468:
465:
464:Jacques Lacan
461:
460:
455:
451:
447:
439:
434:
430:
428:
424:
420:
416:
413:
409:
404:
403:
397:
393:
390:
386:
376:
369:
360:
353:
345:
340:
339:
331:
317:
315:
302:
298:
294:
290:
286:
285:
279:
277:
272:
268:
264:
260:
256:
252:
242:
240:
239:prime numbers
236:
232:
228:
224:
220:
216:
208:
204:
200:
196:
191:
189:
185:
181:
177:
173:
169:
165:
161:
157:
152:
150:
146:
141:
137:
133:
129:
120:
116:
112:
107:
103:
101:
97:
85:
83:
79:
75:
73:
69:
65:
63:
59:
55:
53:
49:
45:
43:
39:
32:
27:
22:
19:
4110:
4098:
4026:Double torus
4011:Braid theory
3826:Crossing no.
3821:Crosscap no.
3615:
3507:Figure-eight
3442:
3403:
3349:
3343:
3337:
3330:Live Science
3329:
3316:
3266:
3260:
3253:
3235:
3231:
3224:
3190:
3184:
3174:
3150:
3144:
3137:
3097:
3091:
3085:
3075:, retrieved
3070:
3061:
3053:
3049:
3039:
3020:
3013:
2973:
2967:
2921:
2917:
2911:
2887:
2882:
2875:
2862:(1): 19â28,
2859:
2855:
2841:
2831:, retrieved
2824:the original
2818:
2769:
2763:
2759:
2755:
2749:
2723:
2676:
2667:
2642:
2636:
2596:math/0103224
2586:
2580:
2563:
2537:math/0402212
2527:
2521:
2469:
2463:
2412:
2406:
2387:, retrieved
2380:the original
2370:(1): 57â60,
2367:
2361:
2345:
2316:
2310:
2304:
2285:
2279:
2270:
2244:
2238:
2225:
2201:
2195:
2189:
2147:
2141:
2135:
2125:
2115:
2107:
2087:
2083:
2079:
2073:
2062:
2054:
2045:
2004:(4): 15â16,
2001:
1995:
1968:
1958:
1942:
1933:
1929:
1921:
1911:
1904:
1880:
1848:
1844:
1808:(5): 41â47,
1805:
1799:
1793:
1769:
1763:
1739:
1733:
1727:
1718:
1712:
1699:
1682:
1673:
1646:
1639:
1629:
1588:(2): 20â25,
1585:
1579:
1573:
1546:
1502:(1): 53â62,
1499:
1493:
1471:
1445:
1407:
1402:
1345:
1311:Efimov state
1308:
1300:halogen bond
1261:
1255:made out of
1242:
1178:RĂ©dei symbol
1159:
1026:
1015:
1000:
922:
846:
840:Logo of the
821:Matthew Cook
818:
798:stick number
781:eccentricity
775:
763:link diagram
744:
678:link diagram
671:
591:
580:
544:
538:
526:
506:
505:" column in
480:
469:
458:
443:
419:Ćmiwa Shrine
415:image stones
389:aristocratic
385:coat of arms
382:
296:
282:
280:
263:Venn diagram
251:link diagram
248:
201:) by linked
192:
184:Efimov state
175:
160:coat of arms
153:
145:Venn diagram
131:
125:
82:AâB notation
42:Crossing no.
18:
4134:Knot theory
3861:Linking no.
3782:Alternating
3583:Conway knot
3563:Carrick mat
3517:Three-twist
3482:Knot theory
3436:Numberphile
3416:Brady Haran
2924:(1): 1â30,
2090:(1): 1â12,
2084:Medium Ăvum
2059:; see also
1928:"The Three
1349:Pronounced
1278:. In 2003,
1257:sheet metal
825:conjectured
796:loops, the
789:icosahedron
737:icosahedron
583:knot theory
536:crescents.
483:knot theory
293:prime links
223:alternating
207:knot theory
199:icosahedron
128:mathematics
115:alternating
56:7.327724753
4128:Categories
4021:Complement
3985:Tabulation
3942:operations
3866:Polynomial
3856:Link group
3851:Knot group
3814:Invertible
3792:Bridge no.
3774:Invariants
3704:Cinquefoil
3573:Perko pair
3499:Hyperbolic
3077:2021-02-12
2833:2012-01-17
2389:2021-03-16
2157:2006.10290
1394:References
1011:hyperbolic
991:complement
849:ropelength
832:Ropelength
704:Ring shape
598:-colorings
558:Linkedness
487:Peter Tait
394:family in
231:hyperbolic
215:-colorings
134:are three
119:hyperbolic
3915:Stick no.
3871:Alexander
3809:Chirality
3754:Solomon's
3714:Septafoil
3641:Satellite
3601:Whitehead
3527:Stevedore
3434:, 2018),
3418:, 2012),
3167:1932-7447
2931:0904.3399
2447:119674622
2422:1406.3370
2288:: 75â98,
2182:219792382
2121:"Mandala"
2018:123311380
1822:120259064
1801:Resonance
1721:: 382â386
1516:189888135
1289:utilised
1104:Λ
1084:…
1078:≈
1055:π
1049:Λ
958:×
905:≈
902:π
876:≈
696:that the
690:2-tangles
633:−
575:2-tangles
427:enneagram
271:alternate
227:algebraic
62:Stick no.
4100:Category
3970:Mutation
3938:Notation
3891:Kauffman
3804:Brunnian
3797:2-bridge
3666:Knot sum
3597:(12n242)
3374:20366816
3301:16541068
3217:28694953
3006:45191675
2998:15166376
2794:64589738
2756:Not Knot
2572:(2002),
2353:(2010),
2233:(1991),
2104:43631234
1951:43490498
1934:Paradiso
1786:22183145
1464:Archived
1444:(eds.),
1436:(2008),
1146:manifold
1017:Not Knot
793:polygons
528:are the
459:Paradiso
392:Borromeo
219:Brunnian
195:ellipses
164:Norsemen
4112:Commons
4031:Fibered
3929:problem
3898:Pretzel
3876:Bracket
3694:Trefoil
3631:L10a140
3591:(11n42)
3585:(11n34)
3553:Endless
3382:7951719
3354:Bibcode
3309:4379828
3281:Bibcode
3208:5477818
3130:4321733
3122:9062186
3102:Bibcode
2978:Bibcode
2969:Science
2940:2605747
2904:2206880
2868:0747855
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