131:
162:
1793:
1005:, where for the category structure, one can compose two tangles if the bottom end of one equals the top end of the other (so the boundaries can be stitched together), by stacking them – they do not literally form a category (pointwise) because there is no identity, since even a trivial tangle takes up vertical space, but up to isotopy they do. The tensor structure is given by juxtaposition of tangles – putting one tangle to the right of the other.
69:
142:
27:
1805:
981:
is a tangle consisting of only intervals, with the ends of each strand required to lie at (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), ... – i.e., connecting the integers, and ending in the same order that they began (one may use any
366:
53:
which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called
847:
520:
1031:
Every link can be cut apart to form a string link, though this is not unique, and invariants of links can sometimes be understood as invariants of string links – this is the case for
852:
says that intervals either connect two lines or connect two points on one of the lines, but imposes no conditions on the circles. One may view tangles as having a vertical direction (
914:
886:
561:
947:
404:
477:
307:
While (1-dimensional) links are defined as embeddings of circles, it is often interesting and especially technically useful to consider embedded intervals (strands), as in
599:
974:) direction. In particular, it must consist solely of intervals, and not double back on itself; however, no specification is made on where on the line the ends lie.
800:
698:
666:
248:
217:
445:
766:
639:
733:
990:-component string link". A string link need not be a braid – it may double back on itself, such as a two-component string link that features an
967:
consists of circles only), braids, and others besides – for example, a strand connecting the two lines together with a circle linked around it.
970:
In this context, a braid is defined as a tangle which is always going down – whose derivative always has a non-zero component in the vertical (
1079:
1020:-component string links, and there is an identity), but not a group, as isotopy classes of string links need not have inverses. However,
1028:
classes) of string links do have inverses, where inverse is given by flipping the string link upside down, and thus form a group.
1170:
1001:
The key technical value of tangles and string links is that they have algebraic structure. Isotopy classes of tangles form a
1738:
324:
1657:
123:
are collectively linked despite the fact that no two of them are directly linked. The
Borromean rings thus form a
1204:
812:
485:
1652:
1647:
1523:
1831:
1809:
1224:
891:
863:
1286:
525:
923:
1356:
1351:
1292:
1163:
374:
1129:
450:
1484:
578:
1698:
1667:
1124:
1032:
65:, but the word is also sometimes used in context where there is no notion of a trivial link.
1528:
1115:
Habegger, Nathan; Masbaum, Gregor (2000), "The
Kontsevich integral and Milnor's invariants",
73:
775:
703:
1797:
1568:
1156:
671:
644:
302:
226:
195:
8:
1605:
1588:
409:
85:
606:
1836:
1626:
1573:
1183:
1098:
50:
1139:
1723:
1672:
1622:
1578:
1538:
1533:
1451:
1002:
111:
The simplest nontrivial example of a link with more than one component is called the
1758:
1583:
1479:
1214:
1134:
1088:
1077:
Habegger, Nathan; Lin, X.S. (1990), "The classification of links up to homotopy",
1718:
1682:
1617:
1563:
1518:
1511:
1401:
1313:
1196:
1048:
278:
120:
97:
89:
31:
1148:
1778:
1677:
1639:
1558:
1471:
1346:
1338:
1298:
20:
130:
1825:
1713:
1501:
1494:
1489:
1036:
991:
141:
124:
1728:
1708:
1612:
1595:
1391:
1328:
957:
308:
134:
101:
1743:
1506:
1411:
1280:
1260:
1250:
1242:
1234:
1179:
81:
54:
42:
39:
603:
Concretely, a connected compact 1-manifold with boundary is an interval
1763:
1748:
1703:
1600:
1553:
1548:
1543:
1373:
1270:
1102:
1058:
995:
920:
and then being able to move in a two-dimensional horizontal direction (
166:
68:
1768:
1436:
269:(considered to be trivially embedded) and a non-trivial embedding of
150:
146:
112:
1093:
1753:
1363:
161:
93:
26:
1773:
1421:
1381:
1053:
285:
is disconnected, the embedding is called a link (or said to be
220:
190:
154:
116:
105:
62:
1662:
177:
The notion of a link can be generalized in a number of ways.
34:, a link with three components each equivalent to the unknot.
16:
Collection of knots which do not intersect, but may be linked
1733:
1016:-component string links form a monoid (one can compose all
277:, non-trivial in the sense that the 2nd embedding is not
219:
diffeomorphic to a disjoint union of a finite number of
952:
between these lines; one can project these to form a
926:
894:
866:
815:
778:
736:
706:
674:
647:
609:
581:
528:
488:
453:
412:
377:
361:{\displaystyle T\colon X\to \mathbf {R} ^{2}\times I}
327:
229:
198:
856:), lying between and possibly connecting two lines
296:
941:
908:
880:
841:
794:
760:
727:
692:
660:
633:
593:
555:
514:
471:
439:
398:
360:
242:
211:
1178:
994:. A braid that is also a string link is called a
57:. Implicit in this definition is that there is a
1823:
1114:
371:of a (smooth) compact 1-manifold with boundary
127:and in fact constitute the simplest such link.
998:, and corresponds with the usual such notion.
1164:
1087:(2), American Mathematical Society: 389–419,
1080:Journal of the American Mathematical Society
836:
824:
541:
529:
509:
497:
261:– the context is that one has a submanifold
1076:
189:is used to describe any submanifold of the
119:) linked together once. The circles in the
1171:
1157:
842:{\displaystyle \mathbf {R} \times \{0,1\}}
515:{\displaystyle \mathbf {R} \times \{0,1\}}
1138:
1128:
1092:
668:(compactness rules out the open interval
982:other fixed set of points); if this has
160:
140:
129:
67:
25:
1824:
1152:
1804:
909:{\displaystyle \mathbf {R} \times 1}
881:{\displaystyle \mathbf {R} \times 0}
257:is essentially the same as the word
180:
115:, which consists of two circles (or
802:The condition that the boundary of
575:together with a fixed embedding of
314:Most generally, one can consider a
293:is connected, it is called a knot.
84:2 link in 3-dimensional space is a
61:reference link, usually called the
13:
582:
556:{\displaystyle \{0,1\}=\partial I}
547:
460:
406:into the plane times the interval
387:
172:
14:
1848:
297:Tangles, string links, and braids
72:A Hopf link spanned by a twisted
1803:
1792:
1791:
942:{\displaystyle \mathbf {R} ^{2}}
929:
896:
868:
817:
490:
342:
1658:Dowker–Thistlethwaite notation
1108:
1070:
755:
743:
719:
707:
687:
675:
628:
616:
466:
457:
431:
419:
399:{\displaystyle (X,\partial X)}
393:
378:
337:
1:
1140:10.1016/S0040-9383(99)00041-5
1064:
1035:, for instance. Compare with
472:{\displaystyle T(\partial X)}
318:– a tangle is an embedding
253:In full generality, the word
571:of a tangle is the manifold
7:
1042:
986:components, we call it an "
700:and the half-open interval
594:{\displaystyle \partial X.}
10:
1853:
963:Tangles include links (if
300:
18:
1787:
1691:
1648:Alexander–Briggs notation
1635:
1470:
1372:
1337:
1195:
19:Not to be confused with
1739:List of knots and links
1287:Kinoshita–Terasaka knot
1024:classes (and thus also
447:such that the boundary
943:
910:
882:
843:
796:
795:{\displaystyle S^{1}.}
762:
729:
728:{\displaystyle [0,1),}
694:
662:
635:
595:
557:
516:
473:
441:
400:
362:
244:
213:
169:
158:
138:
77:
35:
1529:Finite type invariant
944:
911:
883:
844:
797:
763:
730:
695:
693:{\displaystyle (0,1)}
663:
661:{\displaystyle S^{1}}
636:
596:
558:
517:
474:
442:
401:
363:
245:
243:{\displaystyle S^{j}}
214:
212:{\displaystyle S^{n}}
164:
144:
137:linked with a circle.
133:
71:
29:
924:
892:
864:
813:
776:
734:
704:
672:
645:
607:
579:
526:
486:
451:
410:
375:
325:
303:Tangle (mathematics)
227:
196:
185:Frequently the word
98:connected components
1832:Links (knot theory)
1699:Alexander's theorem
1033:Milnor's invariants
1012:isotopy classes of
440:{\displaystyle I=,}
49:is a collection of
939:
906:
878:
839:
792:
761:{\displaystyle I=}
758:
725:
690:
658:
634:{\displaystyle I=}
631:
591:
553:
512:
469:
437:
396:
358:
240:
209:
170:
159:
139:
78:
36:
1819:
1818:
1673:Reidemeister move
1539:Khovanov homology
1534:Hyperbolic volume
956:, analogous to a
181:General manifolds
88:of 3-dimensional
1844:
1807:
1806:
1795:
1794:
1759:Tait conjectures
1462:
1461:
1447:
1446:
1432:
1431:
1324:
1323:
1309:
1308:
1293:(−2,3,7) pretzel
1173:
1166:
1159:
1150:
1149:
1144:
1143:
1142:
1132:
1123:(6): 1253–1289,
1112:
1106:
1105:
1096:
1074:
948:
946:
945:
940:
938:
937:
932:
915:
913:
912:
907:
899:
887:
885:
884:
879:
871:
848:
846:
845:
840:
820:
801:
799:
798:
793:
788:
787:
767:
765:
764:
759:
732:
731:
726:
699:
697:
696:
691:
667:
665:
664:
659:
657:
656:
640:
638:
637:
632:
600:
598:
597:
592:
562:
560:
559:
554:
521:
519:
518:
513:
493:
479:is embedded in
478:
476:
475:
470:
446:
444:
443:
438:
405:
403:
402:
397:
367:
365:
364:
359:
351:
350:
345:
281:to the 1st. If
249:
247:
246:
241:
239:
238:
218:
216:
215:
210:
208:
207:
1852:
1851:
1847:
1846:
1845:
1843:
1842:
1841:
1822:
1821:
1820:
1815:
1783:
1687:
1653:Conway notation
1637:
1631:
1618:Tricolorability
1466:
1460:
1457:
1456:
1455:
1445:
1442:
1441:
1440:
1430:
1427:
1426:
1425:
1417:
1407:
1397:
1387:
1368:
1347:Composite knots
1333:
1322:
1319:
1318:
1317:
1314:Borromean rings
1307:
1304:
1303:
1302:
1276:
1266:
1256:
1246:
1238:
1230:
1220:
1210:
1191:
1177:
1147:
1113:
1109:
1094:10.2307/1990959
1075:
1071:
1067:
1049:Hyperbolic link
1045:
1003:tensor category
933:
928:
927:
925:
922:
921:
895:
893:
890:
889:
867:
865:
862:
861:
816:
814:
811:
810:
783:
779:
777:
774:
773:
735:
705:
702:
701:
673:
670:
669:
652:
648:
646:
643:
642:
608:
605:
604:
580:
577:
576:
527:
524:
523:
489:
487:
484:
483:
452:
449:
448:
411:
408:
407:
376:
373:
372:
346:
341:
340:
326:
323:
322:
305:
299:
234:
230:
228:
225:
224:
203:
199:
197:
194:
193:
183:
175:
173:Generalizations
121:Borromean rings
90:Euclidean space
80:For example, a
32:Borromean rings
24:
17:
12:
11:
5:
1850:
1840:
1839:
1834:
1817:
1816:
1814:
1813:
1801:
1788:
1785:
1784:
1782:
1781:
1779:Surgery theory
1776:
1771:
1766:
1761:
1756:
1751:
1746:
1741:
1736:
1731:
1726:
1721:
1716:
1711:
1706:
1701:
1695:
1693:
1689:
1688:
1686:
1685:
1680:
1678:Skein relation
1675:
1670:
1665:
1660:
1655:
1650:
1644:
1642:
1633:
1632:
1630:
1629:
1623:Unknotting no.
1620:
1615:
1610:
1609:
1608:
1598:
1593:
1592:
1591:
1586:
1581:
1576:
1571:
1561:
1556:
1551:
1546:
1541:
1536:
1531:
1526:
1521:
1516:
1515:
1514:
1504:
1499:
1498:
1497:
1487:
1482:
1476:
1474:
1468:
1467:
1465:
1464:
1458:
1449:
1443:
1434:
1428:
1419:
1415:
1409:
1405:
1399:
1395:
1389:
1385:
1378:
1376:
1370:
1369:
1367:
1366:
1361:
1360:
1359:
1354:
1343:
1341:
1335:
1334:
1332:
1331:
1326:
1320:
1311:
1305:
1296:
1290:
1284:
1278:
1274:
1268:
1264:
1258:
1254:
1248:
1244:
1240:
1236:
1232:
1228:
1222:
1218:
1212:
1208:
1201:
1199:
1193:
1192:
1176:
1175:
1168:
1161:
1153:
1146:
1145:
1130:10.1.1.31.6675
1107:
1068:
1066:
1063:
1062:
1061:
1056:
1051:
1044:
1041:
954:tangle diagram
936:
931:
918:
917:
905:
902:
898:
877:
874:
870:
850:
849:
838:
835:
832:
829:
826:
823:
819:
791:
786:
782:
757:
754:
751:
748:
745:
742:
739:
724:
721:
718:
715:
712:
709:
689:
686:
683:
680:
677:
655:
651:
630:
627:
624:
621:
618:
615:
612:
590:
587:
584:
565:
564:
552:
549:
546:
543:
540:
537:
534:
531:
511:
508:
505:
502:
499:
496:
492:
468:
465:
462:
459:
456:
436:
433:
430:
427:
424:
421:
418:
415:
395:
392:
389:
386:
383:
380:
369:
368:
357:
354:
349:
344:
339:
336:
333:
330:
298:
295:
265:of a manifold
237:
233:
206:
202:
182:
179:
174:
171:
92:(or often the
21:Linking number
15:
9:
6:
4:
3:
2:
1849:
1838:
1835:
1833:
1830:
1829:
1827:
1812:
1811:
1802:
1800:
1799:
1790:
1789:
1786:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1714:Conway sphere
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1696:
1694:
1690:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1645:
1643:
1641:
1634:
1628:
1624:
1621:
1619:
1616:
1614:
1611:
1607:
1604:
1603:
1602:
1599:
1597:
1594:
1590:
1587:
1585:
1582:
1580:
1577:
1575:
1572:
1570:
1567:
1566:
1565:
1562:
1560:
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1513:
1510:
1509:
1508:
1505:
1503:
1500:
1496:
1493:
1492:
1491:
1488:
1486:
1485:Arf invariant
1483:
1481:
1478:
1477:
1475:
1473:
1469:
1453:
1450:
1438:
1435:
1423:
1420:
1413:
1410:
1403:
1400:
1393:
1390:
1383:
1380:
1379:
1377:
1375:
1371:
1365:
1362:
1358:
1355:
1353:
1350:
1349:
1348:
1345:
1344:
1342:
1340:
1336:
1330:
1327:
1315:
1312:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1272:
1269:
1262:
1259:
1252:
1249:
1247:
1241:
1239:
1233:
1226:
1223:
1216:
1213:
1206:
1203:
1202:
1200:
1198:
1194:
1189:
1185:
1181:
1174:
1169:
1167:
1162:
1160:
1155:
1154:
1151:
1141:
1136:
1131:
1126:
1122:
1118:
1111:
1104:
1100:
1095:
1090:
1086:
1082:
1081:
1073:
1069:
1060:
1057:
1055:
1052:
1050:
1047:
1046:
1040:
1038:
1037:closed braids
1034:
1029:
1027:
1023:
1019:
1015:
1011:
1006:
1004:
999:
997:
993:
992:overhand knot
989:
985:
980:
975:
973:
968:
966:
961:
959:
955:
950:
934:
903:
900:
875:
872:
859:
858:
857:
855:
833:
830:
827:
821:
809:
808:
807:
805:
789:
784:
780:
771:
752:
749:
746:
740:
737:
722:
716:
713:
710:
684:
681:
678:
653:
649:
625:
622:
619:
613:
610:
601:
588:
585:
574:
570:
550:
544:
538:
535:
532:
506:
503:
500:
494:
482:
481:
480:
463:
454:
434:
428:
425:
422:
416:
413:
390:
384:
381:
355:
352:
347:
334:
331:
328:
321:
320:
319:
317:
312:
310:
304:
294:
292:
288:
284:
280:
276:
272:
268:
264:
260:
256:
251:
235:
231:
222:
204:
200:
192:
188:
178:
168:
163:
156:
152:
148:
143:
136:
132:
128:
126:
125:Brunnian link
122:
118:
114:
109:
107:
103:
99:
95:
91:
87:
83:
75:
70:
66:
64:
60:
56:
52:
48:
44:
41:
33:
28:
22:
1808:
1796:
1724:Double torus
1709:Braid theory
1524:Crossing no.
1519:Crosscap no.
1205:Figure-eight
1187:
1120:
1116:
1110:
1084:
1078:
1072:
1030:
1025:
1021:
1017:
1013:
1009:
1008:For a fixed
1007:
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987:
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978:
976:
971:
969:
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958:knot diagram
953:
951:
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853:
851:
803:
769:
641:or a circle
602:
572:
568:
566:
370:
315:
313:
309:braid theory
306:
290:
286:
282:
274:
270:
266:
262:
258:
254:
252:
186:
184:
176:
135:Trefoil knot
110:
102:homeomorphic
82:co-dimension
79:
58:
46:
40:mathematical
37:
1559:Linking no.
1480:Alternating
1281:Conway knot
1261:Carrick mat
1215:Three-twist
1180:Knot theory
1022:concordance
979:string link
55:knot theory
43:knot theory
1826:Categories
1719:Complement
1683:Tabulation
1640:operations
1564:Polynomial
1554:Link group
1549:Knot group
1512:Invertible
1490:Bridge no.
1472:Invariants
1402:Cinquefoil
1271:Perko pair
1197:Hyperbolic
1065:References
1059:Link group
996:pure braid
301:See also:
167:torus link
1837:Manifolds
1613:Stick no.
1569:Alexander
1507:Chirality
1452:Solomon's
1412:Septafoil
1339:Satellite
1299:Whitehead
1225:Stevedore
1125:CiteSeerX
901:×
873:×
822:×
806:lies in
583:∂
548:∂
495:×
461:∂
388:∂
353:×
338:→
332::
151:cobordant
147:Hopf link
113:Hopf link
1798:Category
1668:Mutation
1636:Notation
1589:Kauffman
1502:Brunnian
1495:2-bridge
1364:Knot sum
1295:(12n242)
1117:Topology
1043:See also
1026:homotopy
772:circles
279:isotopic
96:) whose
94:3-sphere
86:subspace
1810:Commons
1729:Fibered
1627:problem
1596:Pretzel
1574:Bracket
1392:Trefoil
1329:L10a140
1289:(11n42)
1283:(11n34)
1251:Endless
1103:1990959
289:). If
221:spheres
153:to the
117:unknots
106:circles
74:annulus
59:trivial
1774:Writhe
1744:Ribbon
1579:HOMFLY
1422:Unlink
1382:Unknot
1357:Square
1352:Granny
1127:
1101:
1054:Unlink
316:tangle
287:linked
191:sphere
165:(2,8)
155:unlink
63:unlink
1764:Twist
1749:Slice
1704:Berge
1692:Other
1663:Flype
1601:Prime
1584:Jones
1544:Genus
1374:Torus
1188:links
1184:knots
1099:JSTOR
1083:, 2,
51:knots
1769:Wild
1734:Knot
1638:and
1625:and
1606:list
1437:Hopf
1186:and
888:and
768:and
569:type
567:The
259:knot
255:link
250:.
187:link
145:The
100:are
47:link
45:, a
30:The
1754:Sum
1275:161
1273:(10
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273:in
149:is
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104:to
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988:ℓ
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