148:
180:
164:
28:
196:
132:
1289:
20:
244:
228:
212:
1324:
1301:
591:
358:
299:
104:
76:
1365:
666:
1234:
1153:
147:
441:
179:
163:
700:
153:
195:
1143:
1358:
1148:
1019:
385:
368:
107:
131:
720:
201:
782:
1389:
788:
852:
847:
659:
258:
In general, it is relatively difficult to determine the unknotting number of a given knot. Known cases include:
1384:
1351:
243:
227:
211:
539:
980:
1305:
1194:
1163:
1024:
632:
1293:
1064:
652:
1101:
1084:
457:
Taniyama, Kouki (2009), "Unknotting numbers of diagrams of a given nontrivial knot are unbounded",
308:
1122:
1069:
683:
679:
466:
412:
51:
1219:
1168:
1074:
1034:
1029:
947:
604:
437:
1335:
121:
has unknotting number at least two, and therefore every knot with unknotting number one is a
1254:
1079:
975:
710:
511:
476:
185:
488:
272:
27:
1214:
1178:
1113:
1059:
1014:
1007:
897:
809:
692:
484:
169:
1274:
1173:
1135:
1054:
967:
842:
834:
794:
637:
395:
118:
111:
89:
61:
31:
607:
16:
Minimum number of times a specific knot must be passed through itself to become untied
1378:
1209:
997:
990:
985:
390:
1224:
1204:
1108:
1091:
887:
824:
400:
137:
79:
907:
746:
738:
730:
249:
233:
217:
19:
1331:
1239:
1002:
776:
756:
675:
644:
43:
39:
1259:
1244:
1199:
1096:
1049:
1044:
1039:
869:
766:
529:
480:
364:
302:
263:
122:
434:
The knot book: an elementary introduction to the mathematical theory of knots
1264:
932:
612:
23:
Trefoil knot without 3-fold symmetry being unknotted by one crossing switch.
1323:
125:. The following table show the unknotting numbers for the first few knots:
106:
crossings. The unknotting number of a knot is always less than half of its
1249:
859:
515:
436:. Providence, Rhode Island: American Mathematical Society. p. 56.
54:
is the minimum number of times the knot must be passed through itself (
1269:
917:
877:
502:
Wendt, Hilmar (December 1937). "Die gordische Auflösung von Knoten".
83:
1158:
471:
26:
1229:
648:
371:
have all been determined. (The unknotting number of the 10
1339:
542:
311:
275:
92:
64:
1187:
1131:
966:
868:
833:
691:
585:
352:
293:
98:
70:
58:) to untie it. If a knot has unknotting number
633:Three_Dimensional_Invariants#Unknotting_Number
1359:
660:
8:
459:Journal of Knot Theory and its Ramifications
114:was first defined by Hilmar Wendt in 1936.
1366:
1352:
667:
653:
645:
543:
541:
470:
342:
310:
274:
91:
63:
586:{\displaystyle {\frac {1}{2}}(p-1)(q-1)}
18:
424:
127:
34:being unknotted by undoing one crossing
262:The unknotting number of a nontrivial
7:
1320:
1318:
1300:
82:of the knot which can be changed to
1338:. You can help Knowledge (XXG) by
14:
1322:
1299:
1288:
1287:
242:
226:
210:
194:
178:
162:
146:
130:
380:Other numerical knot invariants
1154:Dowker–Thistlethwaite notation
580:
568:
565:
553:
339:
327:
324:
312:
288:
276:
1:
432:Adams, Colin Conrad (2004).
353:{\displaystyle (p-1)(q-1)/2}
269:The unknotting number of a
1406:
1317:
363:The unknotting numbers of
1283:
1144:Alexander–Briggs notation
504:Mathematische Zeitschrift
481:10.1142/S0218216509007361
1235:List of knots and links
783:Kinoshita–Terasaka knot
375:prime knot is unknown.)
266:is always equal to one.
587:
354:
295:
100:
78:, then there exists a
72:
35:
24:
1025:Finite type invariant
588:
534:Mathworld.Wolfram.com
355:
296:
294:{\displaystyle (p,q)}
101:
73:
30:
22:
540:
309:
273:
90:
62:
1332:knot theory-related
1195:Alexander's theorem
608:"Unknotting Number"
367:with nine or fewer
253:unknotting number 3
237:unknotting number 1
221:unknotting number 1
205:unknotting number 1
189:unknotting number 1
173:unknotting number 2
157:unknotting number 1
141:unknotting number 1
605:Weisstein, Eric W.
583:
516:10.1007/BF01160103
413:Unknotting problem
350:
291:
96:
68:
36:
25:
1390:Knot theory stubs
1347:
1346:
1315:
1314:
1169:Reidemeister move
1035:Khovanov homology
1030:Hyperbolic volume
551:
154:Figure-eight knot
99:{\displaystyle n}
71:{\displaystyle n}
48:unknotting number
1397:
1368:
1361:
1354:
1326:
1319:
1303:
1302:
1291:
1290:
1255:Tait conjectures
958:
957:
943:
942:
928:
927:
820:
819:
805:
804:
789:(−2,3,7) pretzel
669:
662:
655:
646:
619:
618:
617:
600:
594:
592:
590:
589:
584:
552:
544:
526:
520:
519:
499:
493:
491:
474:
465:(8): 1049–1063,
454:
448:
447:
429:
359:
357:
356:
351:
346:
300:
298:
297:
292:
246:
230:
214:
198:
186:Three-twist knot
182:
166:
150:
134:
105:
103:
102:
97:
77:
75:
74:
69:
1405:
1404:
1400:
1399:
1398:
1396:
1395:
1394:
1385:Knot invariants
1375:
1374:
1373:
1372:
1316:
1311:
1279:
1183:
1149:Conway notation
1133:
1127:
1114:Tricolorability
962:
956:
953:
952:
951:
941:
938:
937:
936:
926:
923:
922:
921:
913:
903:
893:
883:
864:
843:Composite knots
829:
818:
815:
814:
813:
810:Borromean rings
803:
800:
799:
798:
772:
762:
752:
742:
734:
726:
716:
706:
687:
673:
628:
623:
622:
603:
602:
601:
597:
538:
537:
527:
523:
501:
500:
496:
456:
455:
451:
444:
431:
430:
426:
421:
409:
386:Crossing number
382:
374:
307:
306:
271:
270:
254:
252:
247:
238:
236:
231:
222:
220:
215:
206:
204:
199:
190:
188:
183:
174:
172:
170:Cinquefoil knot
167:
158:
156:
151:
142:
140:
135:
108:crossing number
88:
87:
60:
59:
56:crossing switch
17:
12:
11:
5:
1403:
1401:
1393:
1392:
1387:
1377:
1376:
1371:
1370:
1363:
1356:
1348:
1345:
1344:
1327:
1313:
1312:
1310:
1309:
1297:
1284:
1281:
1280:
1278:
1277:
1275:Surgery theory
1272:
1267:
1262:
1257:
1252:
1247:
1242:
1237:
1232:
1227:
1222:
1217:
1212:
1207:
1202:
1197:
1191:
1189:
1185:
1184:
1182:
1181:
1176:
1174:Skein relation
1171:
1166:
1161:
1156:
1151:
1146:
1140:
1138:
1129:
1128:
1126:
1125:
1119:Unknotting no.
1116:
1111:
1106:
1105:
1104:
1094:
1089:
1088:
1087:
1082:
1077:
1072:
1067:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1011:
1010:
1000:
995:
994:
993:
983:
978:
972:
970:
964:
963:
961:
960:
954:
945:
939:
930:
924:
915:
911:
905:
901:
895:
891:
885:
881:
874:
872:
866:
865:
863:
862:
857:
856:
855:
850:
839:
837:
831:
830:
828:
827:
822:
816:
807:
801:
792:
786:
780:
774:
770:
764:
760:
754:
750:
744:
740:
736:
732:
728:
724:
718:
714:
708:
704:
697:
695:
689:
688:
674:
672:
671:
664:
657:
649:
643:
642:
638:The Knot Atlas
627:
626:External links
624:
621:
620:
595:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
550:
547:
521:
510:(1): 680–696.
494:
449:
442:
423:
422:
420:
417:
416:
415:
408:
405:
404:
403:
398:
396:Linking number
393:
388:
381:
378:
377:
376:
372:
361:
349:
345:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
290:
287:
284:
281:
278:
267:
256:
255:
248:
241:
239:
232:
225:
223:
216:
209:
207:
202:Stevedore knot
200:
193:
191:
184:
177:
175:
168:
161:
159:
152:
145:
143:
136:
129:
119:composite knot
95:
67:
32:Whitehead link
15:
13:
10:
9:
6:
4:
3:
2:
1402:
1391:
1388:
1386:
1383:
1382:
1380:
1369:
1364:
1362:
1357:
1355:
1350:
1349:
1343:
1341:
1337:
1334:article is a
1333:
1328:
1325:
1321:
1308:
1307:
1298:
1296:
1295:
1286:
1285:
1282:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1210:Conway sphere
1208:
1206:
1203:
1201:
1198:
1196:
1193:
1192:
1190:
1186:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1141:
1139:
1137:
1130:
1124:
1120:
1117:
1115:
1112:
1110:
1107:
1103:
1100:
1099:
1098:
1095:
1093:
1090:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1062:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1009:
1006:
1005:
1004:
1001:
999:
996:
992:
989:
988:
987:
984:
982:
981:Arf invariant
979:
977:
974:
973:
971:
969:
965:
949:
946:
934:
931:
919:
916:
909:
906:
899:
896:
889:
886:
879:
876:
875:
873:
871:
867:
861:
858:
854:
851:
849:
846:
845:
844:
841:
840:
838:
836:
832:
826:
823:
811:
808:
796:
793:
790:
787:
784:
781:
778:
775:
768:
765:
758:
755:
748:
745:
743:
737:
735:
729:
722:
719:
712:
709:
702:
699:
698:
696:
694:
690:
685:
681:
677:
670:
665:
663:
658:
656:
651:
650:
647:
640:
639:
634:
630:
629:
625:
615:
614:
609:
606:
599:
596:
577:
574:
571:
562:
559:
556:
548:
545:
535:
531:
525:
522:
517:
513:
509:
505:
498:
495:
490:
486:
482:
478:
473:
468:
464:
460:
453:
450:
445:
443:0-8218-3678-1
439:
435:
428:
425:
418:
414:
411:
410:
406:
402:
399:
397:
394:
392:
391:Bridge number
389:
387:
384:
383:
379:
370:
366:
362:
347:
343:
336:
333:
330:
321:
318:
315:
304:
285:
282:
279:
268:
265:
261:
260:
259:
251:
245:
240:
235:
229:
224:
219:
213:
208:
203:
197:
192:
187:
181:
176:
171:
165:
160:
155:
149:
144:
139:
133:
128:
126:
124:
120:
115:
113:
109:
93:
86:by switching
85:
81:
65:
57:
53:
49:
45:
41:
33:
29:
21:
1340:expanding it
1329:
1304:
1292:
1220:Double torus
1205:Braid theory
1118:
1020:Crossing no.
1015:Crosscap no.
701:Figure-eight
636:
611:
598:
533:
524:
507:
503:
497:
462:
458:
452:
433:
427:
401:Stick number
305:is equal to
257:
138:Trefoil knot
116:
55:
47:
40:mathematical
37:
1055:Linking no.
976:Alternating
777:Conway knot
757:Carrick mat
711:Three-twist
676:Knot theory
365:prime knots
44:knot theory
1379:Categories
1215:Complement
1179:Tabulation
1136:operations
1060:Polynomial
1050:Link group
1045:Knot group
1008:Invertible
986:Bridge no.
968:Invariants
898:Cinquefoil
767:Perko pair
693:Hyperbolic
530:Torus Knot
419:References
303:torus knot
264:twist knot
123:prime knot
1109:Stick no.
1065:Alexander
1003:Chirality
948:Solomon's
908:Septafoil
835:Satellite
795:Whitehead
721:Stevedore
613:MathWorld
575:−
560:−
472:0805.3174
369:crossings
334:−
319:−
112:invariant
1294:Category
1164:Mutation
1132:Notation
1085:Kauffman
998:Brunnian
991:2-bridge
860:Knot sum
791:(12n242)
407:See also
42:area of
1306:Commons
1225:Fibered
1123:problem
1092:Pretzel
1070:Bracket
888:Trefoil
825:L10a140
785:(11n42)
779:(11n34)
747:Endless
489:2554334
250:7₁ knot
234:6₃ knot
218:6₂ knot
110:. This
80:diagram
38:In the
1270:Writhe
1240:Ribbon
1075:HOMFLY
918:Unlink
878:Unknot
853:Square
848:Granny
487:
440:
84:unknot
46:, the
1330:This
1260:Twist
1245:Slice
1200:Berge
1188:Other
1159:Flype
1097:Prime
1080:Jones
1040:Genus
870:Torus
684:links
680:knots
467:arXiv
50:of a
1336:stub
1265:Wild
1230:Knot
1134:and
1121:and
1102:list
933:Hopf
682:and
438:ISBN
117:Any
52:knot
1250:Sum
771:161
769:(10
635:",
536:. "
532:",
512:doi
477:doi
1381::
950:(4
935:(2
920:(0
910:(7
900:(5
890:(3
880:(0
812:(6
797:(5
761:18
759:(8
749:(7
723:(6
713:(5
703:(4
610:.
593:".
508:42
506:.
485:MR
483:,
475:,
463:18
461:,
373:11
1367:e
1360:t
1353:v
1342:.
959:)
955:1
944:)
940:1
929:)
925:1
914:)
912:1
904:)
902:1
894:)
892:1
884:)
882:1
821:)
817:2
806:)
802:1
773:)
763:)
753:)
751:4
741:3
739:6
733:2
731:6
727:)
725:1
717:)
715:2
707:)
705:1
686:)
678:(
668:e
661:t
654:v
641:.
631:"
616:.
581:)
578:1
572:q
569:(
566:)
563:1
557:p
554:(
549:2
546:1
528:"
518:.
514::
492:.
479::
469::
446:.
360:.
348:2
344:/
340:)
337:1
331:q
328:(
325:)
322:1
316:p
313:(
301:-
289:)
286:q
283:,
280:p
277:(
94:n
66:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
