1276:
1288:
199:
restricts onto a homeomorphism of the complements of the knots, and this restricted homeomorphism induces an isomorphism of fundamental groups. However, it is possible for two inequivalent knots to have isomorphic knot groups (see below for an example).
572:
118:
468:
324:
383:
193:
148:
166:
and can be used to distinguish between certain pairs of inequivalent knots. This is because an equivalence between two knots is a self-homeomorphism of
123:
Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is the fundamental group of its complement in
489:
653:
69:
1221:
412:
271:
1140:
626:
329:
687:
477:
1135:
1130:
1006:
1292:
707:
769:
618:
169:
839:
834:
775:
646:
585:
581:
614:
1314:
967:
260:
1181:
1150:
222:
1011:
221:
The knot group (or fundamental group of an oriented link in general) can be computed in the
1280:
1051:
639:
126:
8:
1088:
1071:
610:
1109:
1056:
670:
666:
24:
1206:
1155:
1105:
1061:
1021:
1016:
934:
622:
48:
1241:
1066:
962:
697:
195:
that is isotopic to the identity and sends the first knot onto the second. Such a
1201:
1165:
1100:
1046:
1001:
994:
884:
796:
679:
52:
36:
631:
1261:
1160:
1122:
1041:
954:
829:
821:
781:
215:
204:
163:
1308:
1196:
984:
977:
972:
196:
1211:
1191:
1095:
1078:
874:
811:
246:
208:
1226:
989:
894:
763:
743:
733:
725:
717:
662:
250:
20:
1246:
1231:
1186:
1083:
1036:
1026:
856:
753:
598:
400:
159:
1251:
919:
28:
588:
have isomorphic knot groups, yet these two knots are not equivalent.
1236:
846:
214:; this follows because the abelianization agrees with the first
1256:
904:
864:
235:
32:
1145:
567:{\displaystyle \langle x,y\mid yxy^{-1}xy=xyx^{-1}yx\rangle }
1216:
113:{\displaystyle \pi _{1}(\mathbb {R} ^{3}\setminus K).}
492:
415:
332:
274:
207:
of a knot group is always isomorphic to the infinite
172:
129:
72:
463:{\displaystyle \langle x,y\mid x^{p}=y^{q}\rangle .}
319:{\displaystyle \langle x,y\mid x^{2}=y^{3}\rangle }
566:
462:
377:
318:
187:
142:
112:
661:
1306:
378:{\displaystyle \langle a,b\mid aba=bab\rangle .}
647:
561:
493:
454:
416:
369:
333:
313:
275:
654:
640:
175:
88:
1307:
16:Fundamental group of a knot complement
635:
1287:
162:knot groups, so the knot group is a
13:
604:
225:by a relatively simple algorithm.
218:, which can be easily computed.
14:
1326:
249:has knot group isomorphic to the
98:
1286:
1275:
1274:
480:has knot group with presentation
403:has knot group with presentation
188:{\displaystyle \mathbb {R} ^{3}}
1141:Dowker–Thistlethwaite notation
104:
83:
1:
238:has knot group isomorphic to
153:
7:
619:Encyclopedia of Mathematics
592:
228:
10:
1331:
158:Two equivalent knots have
1270:
1174:
1131:Alexander–Briggs notation
1118:
953:
855:
820:
678:
1222:List of knots and links
770:Kinoshita–Terasaka knot
568:
464:
379:
320:
223:Wirtinger presentation
189:
144:
114:
1012:Finite type invariant
569:
465:
380:
321:
259:. This group has the
190:
145:
143:{\displaystyle S^{3}}
115:
615:Knot and Link Groups
490:
413:
330:
272:
170:
127:
70:
1182:Alexander's theorem
611:Hazewinkel, Michiel
35:into 3-dimensional
564:
460:
375:
316:
185:
140:
110:
47:is defined as the
1302:
1301:
1156:Reidemeister move
1022:Khovanov homology
1017:Hyperbolic volume
478:figure eight knot
49:fundamental group
1322:
1290:
1289:
1278:
1277:
1242:Tait conjectures
945:
944:
930:
929:
915:
914:
807:
806:
792:
791:
776:(−2,3,7) pretzel
656:
649:
642:
633:
632:
573:
571:
570:
565:
554:
553:
526:
525:
469:
467:
466:
461:
453:
452:
440:
439:
384:
382:
381:
376:
325:
323:
322:
317:
312:
311:
299:
298:
194:
192:
191:
186:
184:
183:
178:
149:
147:
146:
141:
139:
138:
119:
117:
116:
111:
97:
96:
91:
82:
81:
1330:
1329:
1325:
1324:
1323:
1321:
1320:
1319:
1315:Knot invariants
1305:
1304:
1303:
1298:
1266:
1170:
1136:Conway notation
1120:
1114:
1101:Tricolorability
949:
943:
940:
939:
938:
928:
925:
924:
923:
913:
910:
909:
908:
900:
890:
880:
870:
851:
830:Composite knots
816:
805:
802:
801:
800:
797:Borromean rings
790:
787:
786:
785:
759:
749:
739:
729:
721:
713:
703:
693:
674:
660:
613:, ed. (2001), "
607:
605:Further reading
595:
546:
542:
518:
514:
491:
488:
487:
448:
444:
435:
431:
414:
411:
410:
331:
328:
327:
307:
303:
294:
290:
273:
270:
269:
258:
231:
179:
174:
173:
171:
168:
167:
156:
134:
130:
128:
125:
124:
92:
87:
86:
77:
73:
71:
68:
67:
53:knot complement
37:Euclidean space
17:
12:
11:
5:
1328:
1318:
1317:
1300:
1299:
1297:
1296:
1284:
1271:
1268:
1267:
1265:
1264:
1262:Surgery theory
1259:
1254:
1249:
1244:
1239:
1234:
1229:
1224:
1219:
1214:
1209:
1204:
1199:
1194:
1189:
1184:
1178:
1176:
1172:
1171:
1169:
1168:
1163:
1161:Skein relation
1158:
1153:
1148:
1143:
1138:
1133:
1127:
1125:
1116:
1115:
1113:
1112:
1106:Unknotting no.
1103:
1098:
1093:
1092:
1091:
1081:
1076:
1075:
1074:
1069:
1064:
1059:
1054:
1044:
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
999:
998:
997:
987:
982:
981:
980:
970:
965:
959:
957:
951:
950:
948:
947:
941:
932:
926:
917:
911:
902:
898:
892:
888:
882:
878:
872:
868:
861:
859:
853:
852:
850:
849:
844:
843:
842:
837:
826:
824:
818:
817:
815:
814:
809:
803:
794:
788:
779:
773:
767:
761:
757:
751:
747:
741:
737:
731:
727:
723:
719:
715:
711:
705:
701:
695:
691:
684:
682:
676:
675:
659:
658:
651:
644:
636:
630:
629:
627:978-1556080104
606:
603:
602:
601:
594:
591:
590:
589:
577:
576:
575:
574:
563:
560:
557:
552:
549:
545:
541:
538:
535:
532:
529:
524:
521:
517:
513:
510:
507:
504:
501:
498:
495:
482:
481:
473:
472:
471:
470:
459:
456:
451:
447:
443:
438:
434:
430:
427:
424:
421:
418:
405:
404:
388:
387:
386:
385:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
315:
310:
306:
302:
297:
293:
289:
286:
283:
280:
277:
264:
263:
256:
243:
230:
227:
216:homology group
205:abelianization
182:
177:
164:knot invariant
155:
152:
137:
133:
121:
120:
109:
106:
103:
100:
95:
90:
85:
80:
76:
15:
9:
6:
4:
3:
2:
1327:
1316:
1313:
1312:
1310:
1295:
1294:
1285:
1283:
1282:
1273:
1272:
1269:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1238:
1235:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1203:
1200:
1198:
1197:Conway sphere
1195:
1193:
1190:
1188:
1185:
1183:
1180:
1179:
1177:
1173:
1167:
1164:
1162:
1159:
1157:
1154:
1152:
1149:
1147:
1144:
1142:
1139:
1137:
1134:
1132:
1129:
1128:
1126:
1124:
1117:
1111:
1107:
1104:
1102:
1099:
1097:
1094:
1090:
1087:
1086:
1085:
1082:
1080:
1077:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1049:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
996:
993:
992:
991:
988:
986:
983:
979:
976:
975:
974:
971:
969:
968:Arf invariant
966:
964:
961:
960:
958:
956:
952:
936:
933:
921:
918:
906:
903:
896:
893:
886:
883:
876:
873:
866:
863:
862:
860:
858:
854:
848:
845:
841:
838:
836:
833:
832:
831:
828:
827:
825:
823:
819:
813:
810:
798:
795:
783:
780:
777:
774:
771:
768:
765:
762:
755:
752:
745:
742:
735:
732:
730:
724:
722:
716:
709:
706:
699:
696:
689:
686:
685:
683:
681:
677:
672:
668:
664:
657:
652:
650:
645:
643:
638:
637:
634:
628:
624:
620:
616:
612:
609:
608:
600:
597:
596:
587:
583:
579:
578:
558:
555:
550:
547:
543:
539:
536:
533:
530:
527:
522:
519:
515:
511:
508:
505:
502:
499:
496:
486:
485:
484:
483:
479:
475:
474:
457:
449:
445:
441:
436:
432:
428:
425:
422:
419:
409:
408:
407:
406:
402:
398:
394:
390:
389:
372:
366:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
308:
304:
300:
295:
291:
287:
284:
281:
278:
268:
267:
266:
265:
262:
255:
252:
248:
244:
241:
237:
233:
232:
226:
224:
219:
217:
213:
210:
206:
201:
198:
197:homeomorphism
180:
165:
161:
151:
135:
131:
107:
101:
93:
78:
74:
66:
65:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
1291:
1279:
1207:Double torus
1192:Braid theory
1031:
1007:Crossing no.
1002:Crosscap no.
688:Figure-eight
621:, Springer,
396:
392:
261:presentation
253:
247:trefoil knot
239:
220:
211:
209:cyclic group
202:
157:
122:
60:
56:
44:
40:
18:
1042:Linking no.
963:Alternating
764:Conway knot
744:Carrick mat
698:Three-twist
663:Knot theory
586:granny knot
582:square knot
251:braid group
21:mathematics
1202:Complement
1166:Tabulation
1123:operations
1047:Polynomial
1037:Link group
1032:Knot group
995:Invertible
973:Bridge no.
955:Invariants
885:Cinquefoil
754:Perko pair
680:Hyperbolic
599:Link group
401:torus knot
160:isomorphic
154:Properties
43:of a knot
41:knot group
1096:Stick no.
1052:Alexander
990:Chirality
935:Solomon's
895:Septafoil
822:Satellite
782:Whitehead
708:Stevedore
562:⟩
548:−
520:−
506:∣
494:⟨
455:⟩
429:∣
417:⟨
370:⟩
346:∣
334:⟨
314:⟩
288:∣
276:⟨
99:∖
75:π
29:embedding
1309:Category
1281:Category
1151:Mutation
1119:Notation
1072:Kauffman
985:Brunnian
978:2-bridge
847:Knot sum
778:(12n242)
593:See also
584:and the
229:Examples
1293:Commons
1212:Fibered
1110:problem
1079:Pretzel
1057:Bracket
875:Trefoil
812:L10a140
772:(11n42)
766:(11n34)
734:Endless
51:of the
1257:Writhe
1227:Ribbon
1062:HOMFLY
905:Unlink
865:Unknot
840:Square
835:Granny
625:
236:unknot
39:. The
33:circle
27:is an
1247:Twist
1232:Slice
1187:Berge
1175:Other
1146:Flype
1084:Prime
1067:Jones
1027:Genus
857:Torus
671:links
667:knots
31:of a
1252:Wild
1217:Knot
1121:and
1108:and
1089:list
920:Hopf
669:and
623:ISBN
580:The
476:The
245:The
234:The
203:The
25:knot
23:, a
1237:Sum
758:161
756:(10
617:",
391:A (
326:or
59:in
55:of
19:In
1311::
937:(4
922:(2
907:(0
897:(7
887:(5
877:(3
867:(0
799:(6
784:(5
748:18
746:(8
736:(7
710:(6
700:(5
690:(4
399:)-
150:.
63:,
946:)
942:1
931:)
927:1
916:)
912:1
901:)
899:1
891:)
889:1
881:)
879:1
871:)
869:1
808:)
804:2
793:)
789:1
760:)
750:)
740:)
738:4
728:3
726:6
720:2
718:6
714:)
712:1
704:)
702:2
694:)
692:1
673:)
665:(
655:e
648:t
641:v
559:x
556:y
551:1
544:x
540:y
537:x
534:=
531:y
528:x
523:1
516:y
512:x
509:y
503:y
500:,
497:x
458:.
450:q
446:y
442:=
437:p
433:x
426:y
423:,
420:x
397:q
395:,
393:p
373:.
367:b
364:a
361:b
358:=
355:a
352:b
349:a
343:b
340:,
337:a
309:3
305:y
301:=
296:2
292:x
285:y
282:,
279:x
257:3
254:B
242:.
240:Z
212:Z
181:3
176:R
136:3
132:S
108:.
105:)
102:K
94:3
89:R
84:(
79:1
61:R
57:K
45:K
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