1680:
1692:
24:
364:: it is not possible to color the strands of its diagram with three colors, so that at least two of the colors are used and so that every crossing has one or three colors present. Each link has only one strand, and if both strands are given the same color then only one color is used, while if they are given different colors then the crossings will have two colors present.
384:, with the property that the inverse image of each point on the 2-sphere is a circle. Thus, these images decompose the 3-sphere into a continuous family of circles, and each two distinct circles form a Hopf link. This was Hopf's motivation for studying the Hopf link: because each two fibers are linked, the Hopf fibration is a nontrivial
895:
Dabrowski-Tumanski, Pawel; Jarmolinska, Aleksandra I.; Niemyska, Wanda; Rawdon, Eric J.; Millett, Kenneth C.; Sulkowska, Joanna I. (2017-01-04), "LinkProt: a database collecting information about biological links",
180:
417:
300:
1738:
1057:
1723:
1625:
138:
1544:
814:
787:
760:
717:
567:
495:
562:, Translations of Mathematical Monographs, vol. 154, Providence, RI: American Mathematical Society, p. 6,
1733:
592:
515:
1091:
1743:
1728:
1539:
1534:
1410:
101:
51:
1696:
1111:
1173:
833:
Dabrowski-Tumanski, Pawel; Sulkowska, Joanna I. (2017-03-28), "Topological knots and links in proteins",
269:
559:
Knots, links, braids and 3-manifolds: An introduction to the new invariants in low-dimensional topology
400:
The Hopf link is also present in some proteins. It consists of two covalent loops, formed by pieces of
389:
439:
before the work of Hopf. It has also long been used outside mathematics, for instance as the crest of
1243:
1238:
1179:
1050:
330:
110:
1371:
225:
in perpendicular planes, each passing through the center of the other. This model minimizes the
1585:
1554:
246:
804:
750:
736:
707:
485:
1415:
957:
952:
777:
693:
540:
128:
557:
1718:
1684:
1455:
1043:
842:
676:
657:
634:
614:
577:
535:
440:
436:
420:
229:
of the link and until 2002 the Hopf link was the only link whose ropelength was known. The
8:
1492:
1475:
712:, Annals of Mathematics Studies, vol. 115, Princeton University Press, p. 373,
380:(a three-dimensional surface in four-dimensional Euclidean space) into the more familiar
322:
846:
618:
1513:
1460:
1074:
1070:
978:
926:
873:
638:
604:
350:
202:
408:. The Hopf link topology is highly conserved in proteins and adds to their stability.
1610:
1559:
1509:
1465:
1425:
1420:
1338:
1000:
982:
931:
913:
878:
860:
810:
783:
756:
713:
563:
491:
464:
342:
91:
61:
1645:
1470:
1366:
1101:
970:
921:
905:
868:
850:
755:, De Gruyter studies in mathematics, vol. 18, Walter de Gruyter, p. 194,
622:
523:
353:
on two generators), distinguishing it from an unlinked pair of loops which has the
163:
642:
1605:
1569:
1504:
1450:
1405:
1398:
1288:
1200:
1083:
966:
672:
630:
573:
531:
452:
334:
306:
1035:
527:
179:
1665:
1564:
1526:
1445:
1358:
1233:
1225:
1185:
1023:
432:
373:
250:
183:
71:
1004:
626:
1712:
1600:
1388:
1381:
1376:
917:
864:
487:
The Knot Book: An
Elementary Introduction to the Mathematical Theory of Knots
855:
1615:
1595:
1499:
1482:
1278:
1215:
935:
894:
882:
522:, IMA Vol. Math. Appl., vol. 103, New York: Springer, pp. 67–78,
401:
361:
171:
81:
41:
31:
1630:
1393:
1298:
1167:
1147:
1137:
1129:
1121:
1066:
909:
230:
222:
194:
191:
1650:
1635:
1590:
1487:
1440:
1435:
1430:
1260:
1157:
974:
953:"Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche"
948:
428:
416:
354:
338:
261:
257:
226:
210:
167:
1655:
1009:
609:
405:
385:
151:
1640:
1250:
458:
377:
1030:
147:
520:
Topology and geometry in polymer science (Minneapolis, MN, 1996)
1660:
1308:
1268:
381:
206:
1549:
326:
234:
23:
1620:
832:
431:, who considered it in 1931 as part of his research on the
1018:
595:(2002), "On the minimum ropelength of knots and links",
590:
443:, a Japanese Buddhist sect founded in the 16th century.
272:
998:
294:
205:with more than one component. It consists of two
1065:
1031:"LinkProt" - the database of known protein links.
555:
209:linked together exactly once, and is named after
1710:
518:(1998), "On distortion and thickness of knots",
835:Proceedings of the National Academy of Sciences
655:
513:
490:, American Mathematical Society, p. 151,
1051:
233:of these two circles forms a shape called an
752:Quantum Invariants of Knots and 3-manifolds
435:. However, in mathematics, it was known to
1058:
1044:
656:Dirnböck, Hans; Stachel, Hellmuth (1997),
556:Prasolov, V. V.; Sossinsky, A. B. (1997),
22:
925:
872:
854:
608:
291:
705:
427:The Hopf link is named after topologist
415:
216:
178:
802:
775:
1711:
748:
591:Cantarella, Jason; Kusner, Robert B.;
551:
549:
509:
507:
1039:
999:
828:
826:
732:
689:
483:
1691:
947:
546:
504:
467:, two loops which are doubly linked
13:
823:
461:, a molecule with two linked loops
388:. This example began the study of
376:is a continuous function from the
295:{\displaystyle \sigma _{1}^{2}.\,}
14:
1755:
992:
665:Journal for Geometry and Graphics
221:A concrete model consists of two
1739:Non-tricolorable knots and links
1690:
1679:
1678:
455:, a link with three closed loops
357:on two generators as its group.
941:
888:
796:
769:
1545:Dowker–Thistlethwaite notation
742:
726:
699:
683:
658:"The development of the oloid"
649:
584:
477:
367:
1:
471:
240:
16:Simplest nontrivial knot link
749:Turaev, Vladimir G. (2010),
484:Adams, Colin Conrad (2004),
333:, so the Hopf link is not a
7:
1724:Alternating knots and links
706:Kauffman, Louis H. (1987),
528:10.1007/978-1-4612-1712-1_7
446:
201:is the simplest nontrivial
10:
1760:
809:, CRC Press, p. 368,
803:Shastri, Anant R. (2013),
411:
395:
390:homotopy groups of spheres
331:locally Euclidean geometry
249:of the two components the
245:Depending on the relative
1674:
1578:
1535:Alexander–Briggs notation
1522:
1357:
1259:
1224:
1082:
627:10.1007/s00222-002-0234-y
256:The Hopf link is a (2,2)-
162:
157:
137:
127:
109:
100:
90:
80:
70:
60:
50:
40:
30:
21:
806:Basic Algebraic Topology
597:Inventiones Mathematicae
253:of the Hopf link is ±1.
1734:Fibered knots and links
1626:List of knots and links
1174:Kinoshita–Terasaka knot
856:10.1073/pnas.1615862114
776:Hatcher, Allen (2002),
898:Nucleic Acids Research
424:
345:of its complement) is
341:of the Hopf link (the
296:
187:
1744:Prime knots and links
1729:Torus knots and links
1416:Finite type invariant
958:Mathematische Annalen
419:
360:The Hopf-link is not
297:
217:Geometric realization
182:
539:. See in particular
437:Carl Friedrich Gauss
309:of the Hopf link is
270:
1586:Alexander's theorem
847:2017PNAS..114.3415D
619:2002InMat.150..257C
514:Kusner, Robert B.;
329:. This space has a
287:
1001:Weisstein, Eric W.
975:10.1007/BF01457962
910:10.1093/nar/gkw976
779:Algebraic Topology
425:
351:free abelian group
317: ×
313: ×
292:
273:
188:
186:for the Hopf link.
1706:
1705:
1560:Reidemeister move
1426:Khovanov homology
1421:Hyperbolic volume
904:(D1): D243–D249,
841:(13): 3415–3420,
735:, Exercise 5.22,
593:Sullivan, John M.
516:Sullivan, John M.
343:fundamental group
177:
176:
62:Hyperbolic volume
1751:
1694:
1693:
1682:
1681:
1646:Tait conjectures
1349:
1348:
1334:
1333:
1319:
1318:
1211:
1210:
1196:
1195:
1180:(−2,3,7) pretzel
1060:
1053:
1046:
1037:
1036:
1014:
1013:
987:
985:
945:
939:
938:
929:
892:
886:
885:
876:
858:
830:
821:
819:
800:
794:
792:
773:
767:
765:
746:
740:
730:
724:
722:
703:
697:
687:
681:
679:
662:
653:
647:
645:
612:
588:
582:
580:
553:
544:
538:
511:
502:
500:
481:
402:protein backbone
301:
299:
298:
293:
286:
281:
142:
123:
122:
26:
19:
18:
1759:
1758:
1754:
1753:
1752:
1750:
1749:
1748:
1709:
1708:
1707:
1702:
1670:
1574:
1540:Conway notation
1524:
1518:
1505:Tricolorability
1353:
1347:
1344:
1343:
1342:
1332:
1329:
1328:
1327:
1317:
1314:
1313:
1312:
1304:
1294:
1284:
1274:
1255:
1234:Composite knots
1220:
1209:
1206:
1205:
1204:
1201:Borromean rings
1194:
1191:
1190:
1189:
1163:
1153:
1143:
1133:
1125:
1117:
1107:
1097:
1078:
1064:
995:
990:
946:
942:
893:
889:
831:
824:
817:
801:
797:
790:
774:
770:
763:
747:
743:
731:
727:
720:
704:
700:
688:
684:
660:
654:
650:
589:
585:
570:
554:
547:
512:
505:
498:
482:
478:
474:
453:Borromean rings
449:
414:
406:disulfide bonds
398:
370:
335:hyperbolic link
307:knot complement
282:
277:
271:
268:
267:
243:
219:
140:
121:
118:
117:
116:
102:Conway notation
17:
12:
11:
5:
1757:
1747:
1746:
1741:
1736:
1731:
1726:
1721:
1704:
1703:
1701:
1700:
1688:
1675:
1672:
1671:
1669:
1668:
1666:Surgery theory
1663:
1658:
1653:
1648:
1643:
1638:
1633:
1628:
1623:
1618:
1613:
1608:
1603:
1598:
1593:
1588:
1582:
1580:
1576:
1575:
1573:
1572:
1567:
1565:Skein relation
1562:
1557:
1552:
1547:
1542:
1537:
1531:
1529:
1520:
1519:
1517:
1516:
1510:Unknotting no.
1507:
1502:
1497:
1496:
1495:
1485:
1480:
1479:
1478:
1473:
1468:
1463:
1458:
1448:
1443:
1438:
1433:
1428:
1423:
1418:
1413:
1408:
1403:
1402:
1401:
1391:
1386:
1385:
1384:
1374:
1369:
1363:
1361:
1355:
1354:
1352:
1351:
1345:
1336:
1330:
1321:
1315:
1306:
1302:
1296:
1292:
1286:
1282:
1276:
1272:
1265:
1263:
1257:
1256:
1254:
1253:
1248:
1247:
1246:
1241:
1230:
1228:
1222:
1221:
1219:
1218:
1213:
1207:
1198:
1192:
1183:
1177:
1171:
1165:
1161:
1155:
1151:
1145:
1141:
1135:
1131:
1127:
1123:
1119:
1115:
1109:
1105:
1099:
1095:
1088:
1086:
1080:
1079:
1063:
1062:
1055:
1048:
1040:
1034:
1033:
1028:
1024:The Knot Atlas
1015:
994:
993:External links
991:
989:
988:
940:
887:
822:
815:
795:
788:
782:, p. 24,
768:
761:
741:
725:
718:
698:
682:
671:(2): 105–118,
648:
603:(2): 257–286,
583:
568:
545:
503:
496:
475:
473:
470:
469:
468:
465:Solomon's knot
462:
456:
448:
445:
433:Hopf fibration
413:
410:
404:, closed with
397:
394:
374:Hopf fibration
369:
366:
303:
302:
290:
285:
280:
276:
251:linking number
242:
239:
218:
215:
184:Skein relation
175:
174:
160:
159:
155:
154:
145:
135:
134:
131:
129:Thistlethwaite
125:
124:
119:
113:
107:
106:
104:
98:
97:
94:
92:Unknotting no.
88:
87:
84:
78:
77:
74:
68:
67:
64:
58:
57:
54:
48:
47:
44:
38:
37:
34:
28:
27:
15:
9:
6:
4:
3:
2:
1756:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1716:
1714:
1699:
1698:
1689:
1687:
1686:
1677:
1676:
1673:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1632:
1629:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1604:
1602:
1601:Conway sphere
1599:
1597:
1594:
1592:
1589:
1587:
1584:
1583:
1581:
1577:
1571:
1568:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1532:
1530:
1528:
1521:
1515:
1511:
1508:
1506:
1503:
1501:
1498:
1494:
1491:
1490:
1489:
1486:
1484:
1481:
1477:
1474:
1472:
1469:
1467:
1464:
1462:
1459:
1457:
1454:
1453:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1400:
1397:
1396:
1395:
1392:
1390:
1387:
1383:
1380:
1379:
1378:
1375:
1373:
1372:Arf invariant
1370:
1368:
1365:
1364:
1362:
1360:
1356:
1340:
1337:
1325:
1322:
1310:
1307:
1300:
1297:
1290:
1287:
1280:
1277:
1270:
1267:
1266:
1264:
1262:
1258:
1252:
1249:
1245:
1242:
1240:
1237:
1236:
1235:
1232:
1231:
1229:
1227:
1223:
1217:
1214:
1202:
1199:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1159:
1156:
1149:
1146:
1139:
1136:
1134:
1128:
1126:
1120:
1113:
1110:
1103:
1100:
1093:
1090:
1089:
1087:
1085:
1081:
1076:
1072:
1068:
1061:
1056:
1054:
1049:
1047:
1042:
1041:
1038:
1032:
1029:
1026:
1025:
1020:
1016:
1012:
1011:
1006:
1002:
997:
996:
984:
980:
976:
972:
968:
965:(1), Berlin:
964:
960:
959:
954:
950:
944:
937:
933:
928:
923:
919:
915:
911:
907:
903:
899:
891:
884:
880:
875:
870:
866:
862:
857:
852:
848:
844:
840:
836:
829:
827:
818:
816:9781466562431
812:
808:
807:
799:
791:
789:9787302105886
785:
781:
780:
772:
764:
762:9783110221831
758:
754:
753:
745:
738:
734:
729:
721:
719:9780691084350
715:
711:
710:
702:
695:
691:
686:
678:
674:
670:
666:
659:
652:
644:
640:
636:
632:
628:
624:
620:
616:
611:
606:
602:
598:
594:
587:
579:
575:
571:
569:0-8218-0588-6
565:
561:
560:
552:
550:
542:
537:
533:
529:
525:
521:
517:
510:
508:
499:
497:9780821836781
493:
489:
488:
480:
476:
466:
463:
460:
457:
454:
451:
450:
444:
442:
438:
434:
430:
422:
418:
409:
407:
403:
393:
391:
387:
383:
379:
375:
365:
363:
358:
356:
352:
348:
344:
340:
336:
332:
328:
324:
320:
316:
312:
308:
288:
283:
278:
274:
266:
265:
264:
263:
259:
254:
252:
248:
238:
236:
232:
228:
224:
214:
212:
208:
204:
200:
196:
193:
185:
181:
173:
169:
165:
161:
156:
153:
149:
146:
144:
136:
132:
130:
126:
114:
112:
108:
105:
103:
99:
95:
93:
89:
85:
83:
79:
75:
73:
69:
65:
63:
59:
55:
53:
49:
45:
43:
39:
35:
33:
29:
25:
20:
1695:
1683:
1611:Double torus
1596:Braid theory
1411:Crossing no.
1406:Crosscap no.
1323:
1092:Figure-eight
1022:
1008:
962:
956:
943:
901:
897:
890:
838:
834:
805:
798:
778:
771:
751:
744:
733:Adams (2004)
728:
708:
701:
690:Adams (2004)
685:
668:
664:
651:
610:math/0103224
600:
596:
586:
558:
519:
486:
479:
426:
399:
371:
362:tricolorable
359:
346:
318:
314:
310:
304:
255:
247:orientations
244:
223:unit circles
220:
198:
192:mathematical
189:
111:A–B notation
52:Crossing no.
32:Braid length
1719:Knot theory
1446:Linking no.
1367:Alternating
1168:Conway knot
1148:Carrick mat
1102:Three-twist
1067:Knot theory
1005:"Hopf Link"
969:: 637–665,
949:Hopf, Heinz
737:p. 133
368:Hopf bundle
231:convex hull
195:knot theory
164:alternating
72:Linking no.
1713:Categories
1606:Complement
1570:Tabulation
1527:operations
1451:Polynomial
1441:Link group
1436:Knot group
1399:Invertible
1377:Bridge no.
1359:Invariants
1289:Cinquefoil
1158:Perko pair
1084:Hyperbolic
694:p. 21
541:p. 77
472:References
429:Heinz Hopf
355:free group
339:knot group
262:braid word
258:torus link
241:Properties
227:ropelength
211:Heinz Hopf
1500:Stick no.
1456:Alexander
1394:Chirality
1339:Solomon's
1299:Septafoil
1226:Satellite
1186:Whitehead
1112:Stevedore
1019:Hopf link
1010:MathWorld
983:123533891
918:0305-1048
865:0027-8424
386:fibration
275:σ
260:with the
199:Hopf link
82:Stick no.
42:Braid no.
1685:Category
1555:Mutation
1523:Notation
1476:Kauffman
1389:Brunnian
1382:2-bridge
1251:Knot sum
1182:(12n242)
967:Springer
951:(1931),
936:27794552
883:28280100
709:On Knots
459:Catenane
447:See also
441:Buzan-ha
421:Buzan-ha
382:2-sphere
378:3-sphere
323:cylinder
1697:Commons
1616:Fibered
1514:problem
1483:Pretzel
1461:Bracket
1279:Trefoil
1216:L10a140
1176:(11n42)
1170:(11n34)
1138:Endless
927:5210653
874:5380043
843:Bibcode
677:1622664
635:1933586
615:Bibcode
578:1414898
536:1655037
412:History
396:Biology
325:over a
207:circles
172:fibered
150:/
1661:Writhe
1631:Ribbon
1466:HOMFLY
1309:Unlink
1269:Unknot
1244:Square
1239:Granny
981:
934:
924:
916:
881:
871:
863:
813:
786:
759:
716:
675:
643:730891
641:
633:
576:
566:
534:
494:
337:. The
321:, the
197:, the
170:,
166:,
141:
139:Last /
1651:Twist
1636:Slice
1591:Berge
1579:Other
1550:Flype
1488:Prime
1471:Jones
1431:Genus
1261:Torus
1075:links
1071:knots
979:S2CID
661:(PDF)
639:S2CID
605:arXiv
423:crest
349:(the
327:torus
235:oloid
168:torus
158:Other
1656:Wild
1621:Knot
1525:and
1512:and
1493:list
1324:Hopf
1073:and
932:PMID
914:ISSN
879:PMID
861:ISSN
811:ISBN
784:ISBN
757:ISBN
714:ISBN
564:ISBN
492:ISBN
372:The
305:The
203:link
152:L4a1
143:Next
133:L2a1
1641:Sum
1162:161
1160:(10
1021:",
971:doi
963:104
922:PMC
906:doi
869:PMC
851:doi
839:114
623:doi
601:150
524:doi
190:In
1715::
1341:(4
1326:(2
1311:(0
1301:(7
1291:(5
1281:(3
1271:(0
1203:(6
1188:(5
1152:18
1150:(8
1140:(7
1114:(6
1104:(5
1094:(4
1007:,
1003:,
977:,
961:,
955:,
930:,
920:,
912:,
902:45
900:,
877:,
867:,
859:,
849:,
837:,
825:^
692:,
673:MR
667:,
663:,
637:,
631:MR
629:,
621:,
613:,
599:,
574:MR
572:,
548:^
532:MR
530:,
506:^
392:.
237:.
213:.
148:L0
1350:)
1346:1
1335:)
1331:1
1320:)
1316:1
1305:)
1303:1
1295:)
1293:1
1285:)
1283:1
1275:)
1273:1
1212:)
1208:2
1197:)
1193:1
1164:)
1154:)
1144:)
1142:4
1132:3
1130:6
1124:2
1122:6
1118:)
1116:1
1108:)
1106:2
1098:)
1096:1
1077:)
1069:(
1059:e
1052:t
1045:v
1027:.
1017:"
986:.
973::
908::
853::
845::
820:.
793:.
766:.
739:.
723:.
696:.
680:.
669:1
646:.
625::
617::
607::
581:.
543:.
526::
501:.
347:Z
319:S
315:S
311:R
289:.
284:2
279:1
120:1
115:2
96:1
86:6
76:1
66:0
56:2
46:2
36:2
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
