1168:
421:
1180:
31:
148:
138:
545:
1113:
1032:
579:
114:
1211:
1022:
1027:
898:
425:
128:
106:
1216:
599:
661:
667:
731:
726:
538:
1206:
859:
410:
1184:
1073:
1042:
903:
1172:
943:
531:
980:
963:
457:
1001:
948:
562:
558:
53:
1098:
1047:
997:
953:
913:
908:
826:
493:
1133:
958:
854:
589:
511:
1093:
1057:
992:
938:
893:
886:
776:
688:
571:
1153:
1052:
1014:
933:
846:
721:
713:
673:
516:
441:
1200:
1088:
876:
869:
864:
445:
72:. It can be a nontrivial problem to determine whether a given knot is prime or not.
1103:
1083:
987:
970:
766:
703:
414:
110:
786:
625:
617:
609:
413:
are counted only once in this table and the following chart (i.e. a knot and its
27:
Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
1118:
881:
655:
635:
554:
523:
41:
496:
428:, not including mirror-images, plus the unknot (which is not considered prime).
1138:
1123:
1078:
928:
923:
918:
748:
645:
76:
1143:
811:
501:
113:
with three crossings. The trefoil is actually a (2, 3)-torus knot. The
34:
17:
1128:
738:
475:
Schubert, H. "Die eindeutige
Zerlegbarkeit eines Knotens in Primknoten".
61:
117:, with four crossings, is the simplest non-torus knot. For any positive
420:
118:
99:
56:
that is, in a certain sense, indecomposable. Specifically, it is a non-
30:
1148:
796:
756:
57:
64:
of two non-trivial knots. Knots that are not prime are said to be
1037:
444:(1919-2001) states that every knot can be uniquely expressed as a
419:
80:
1108:
527:
348:
143:
133:
1066:
1010:
845:
747:
712:
570:
79:. These are formed by wrapping a circle around a
124:, there are a finite number of prime knots with
424:A chart of all prime knots with seven or fewer
539:
8:
477:S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl.
75:A family of examples of prime knots are the
546:
532:
524:
155:
29:
468:
512:Prime Links with a Non-Prime Component
7:
1179:
151:) are given in the following table.
60:knot which cannot be written as the
25:
131:. The first few values (sequence
109:. The simplest prime knot is the
105:Knots are characterized by their
1178:
1167:
1166:
1033:Dowker–Thistlethwaite notation
1:
417:are considered equivalent).
86:times in one direction and
1233:
479:1949 (1949), 57–104.
90:times in the other, where
1162:
1023:Alexander–Briggs notation
1114:List of knots and links
662:Kinoshita–Terasaka knot
435:
429:
37:
1212:Prime knots and links
904:Finite type invariant
423:
223:Number of prime knots
33:
1217:Knots (knot theory)
1074:Alexander's theorem
458:List of prime knots
35:Simplest prime link
494:Weisstein, Eric W.
436:Schubert's theorem
430:
38:
1194:
1193:
1048:Reidemeister move
914:Khovanov homology
909:Hyperbolic volume
440:A theorem due to
406:
405:
115:figure-eight knot
16:(Redirected from
1224:
1182:
1181:
1170:
1169:
1134:Tait conjectures
837:
836:
822:
821:
807:
806:
699:
698:
684:
683:
668:(−2,3,7) pretzel
548:
541:
534:
525:
507:
506:
480:
473:
448:of prime knots.
291:Composite knots
156:
146:
141:) and (sequence
136:
107:crossing numbers
21:
1232:
1231:
1227:
1226:
1225:
1223:
1222:
1221:
1207:Knot invariants
1197:
1196:
1195:
1190:
1158:
1062:
1028:Conway notation
1012:
1006:
993:Tricolorability
841:
835:
832:
831:
830:
820:
817:
816:
815:
805:
802:
801:
800:
792:
782:
772:
762:
743:
722:Composite knots
708:
697:
694:
693:
692:
689:Borromean rings
682:
679:
678:
677:
651:
641:
631:
621:
613:
605:
595:
585:
566:
552:
492:
491:
488:
483:
474:
470:
466:
454:
438:
432:
224:
142:
132:
70:composite links
66:composite knots
28:
23:
22:
15:
12:
11:
5:
1230:
1228:
1220:
1219:
1214:
1209:
1199:
1198:
1192:
1191:
1189:
1188:
1176:
1163:
1160:
1159:
1157:
1156:
1154:Surgery theory
1151:
1146:
1141:
1136:
1131:
1126:
1121:
1116:
1111:
1106:
1101:
1096:
1091:
1086:
1081:
1076:
1070:
1068:
1064:
1063:
1061:
1060:
1055:
1053:Skein relation
1050:
1045:
1040:
1035:
1030:
1025:
1019:
1017:
1008:
1007:
1005:
1004:
998:Unknotting no.
995:
990:
985:
984:
983:
973:
968:
967:
966:
961:
956:
951:
946:
936:
931:
926:
921:
916:
911:
906:
901:
896:
891:
890:
889:
879:
874:
873:
872:
862:
857:
851:
849:
843:
842:
840:
839:
833:
824:
818:
809:
803:
794:
790:
784:
780:
774:
770:
764:
760:
753:
751:
745:
744:
742:
741:
736:
735:
734:
729:
718:
716:
710:
709:
707:
706:
701:
695:
686:
680:
671:
665:
659:
653:
649:
643:
639:
633:
629:
623:
619:
615:
611:
607:
603:
597:
593:
587:
583:
576:
574:
568:
567:
553:
551:
550:
543:
536:
528:
522:
521:
517:The Knot Atlas
508:
487:
486:External links
484:
482:
481:
467:
465:
462:
461:
460:
453:
450:
442:Horst Schubert
437:
434:
408:
407:
404:
403:
400:
398:
395:
393:
390:
388:
385:
383:
380:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
345:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
288:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
230:
220:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
177:
174:
171:
168:
165:
162:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1229:
1218:
1215:
1213:
1210:
1208:
1205:
1204:
1202:
1187:
1186:
1177:
1175:
1174:
1165:
1164:
1161:
1155:
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1089:Conway sphere
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1071:
1069:
1065:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1020:
1018:
1016:
1009:
1003:
999:
996:
994:
991:
989:
986:
982:
979:
978:
977:
974:
972:
969:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
941:
940:
937:
935:
932:
930:
927:
925:
922:
920:
917:
915:
912:
910:
907:
905:
902:
900:
897:
895:
892:
888:
885:
884:
883:
880:
878:
875:
871:
868:
867:
866:
863:
861:
860:Arf invariant
858:
856:
853:
852:
850:
848:
844:
828:
825:
813:
810:
798:
795:
788:
785:
778:
775:
768:
765:
758:
755:
754:
752:
750:
746:
740:
737:
733:
730:
728:
725:
724:
723:
720:
719:
717:
715:
711:
705:
702:
690:
687:
675:
672:
669:
666:
663:
660:
657:
654:
647:
644:
637:
634:
627:
624:
622:
616:
614:
608:
601:
598:
591:
588:
581:
578:
577:
575:
573:
569:
564:
560:
556:
549:
544:
542:
537:
535:
530:
529:
526:
519:
518:
513:
509:
504:
503:
498:
495:
490:
489:
485:
478:
472:
469:
463:
459:
456:
455:
451:
449:
447:
446:connected sum
443:
433:
427:
422:
418:
416:
412:
411:Enantiomorphs
401:
399:
396:
394:
391:
389:
386:
384:
381:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
350:
347:
346:
342:
340:
337:
335:
332:
330:
327:
325:
322:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
289:
285:
282:
279:
276:
273:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
240:
237:
234:
231:
228:
222:
221:
217:
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
166:
163:
161:
158:
157:
154:
153:
152:
150:
145:
140:
135:
130:
127:
123:
120:
116:
112:
108:
103:
101:
97:
93:
89:
85:
82:
78:
73:
71:
67:
63:
59:
55:
51:
47:
43:
36:
32:
19:
1183:
1171:
1099:Double torus
1084:Braid theory
975:
899:Crossing no.
894:Crosscap no.
580:Figure-eight
515:
500:
497:"Prime Knot"
476:
471:
439:
431:
415:mirror image
409:
226:
159:
125:
121:
104:
95:
91:
87:
83:
74:
69:
65:
49:
45:
39:
934:Linking no.
855:Alternating
656:Conway knot
636:Carrick mat
590:Three-twist
555:Knot theory
77:torus knots
42:knot theory
18:Prime knots
1201:Categories
1094:Complement
1058:Tabulation
1015:operations
939:Polynomial
929:Link group
924:Knot group
887:Invertible
865:Bridge no.
847:Invariants
777:Cinquefoil
646:Perko pair
572:Hyperbolic
464:References
286:294130458
229:crossings
102:integers.
50:prime link
46:prime knot
988:Stick no.
944:Alexander
882:Chirality
827:Solomon's
787:Septafoil
714:Satellite
674:Whitehead
600:Stevedore
502:MathWorld
426:crossings
129:crossings
1173:Category
1043:Mutation
1011:Notation
964:Kauffman
877:Brunnian
870:2-bridge
739:Knot sum
670:(12n242)
452:See also
283:48266466
62:knot sum
1185:Commons
1104:Fibered
1002:problem
971:Pretzel
949:Bracket
767:Trefoil
704:L10a140
664:(11n42)
658:(11n34)
626:Endless
280:8053393
277:1388705
147:in the
144:A086825
137:in the
134:A002863
119:integer
111:trefoil
100:coprime
58:trivial
1149:Writhe
1119:Ribbon
954:HOMFLY
797:Unlink
757:Unknot
732:Square
727:Granny
274:253293
1139:Twist
1124:Slice
1079:Berge
1067:Other
1038:Flype
976:Prime
959:Jones
919:Genus
749:Torus
563:links
559:knots
349:Total
271:46972
225:with
81:torus
52:is a
1144:Wild
1109:Knot
1013:and
1000:and
981:list
812:Hopf
561:and
402:...
343:...
268:9988
265:2176
149:OEIS
139:OEIS
98:are
94:and
54:knot
44:, a
1129:Sum
650:161
648:(10
514:",
397:...
392:...
387:...
382:...
377:...
338:...
333:...
328:...
323:...
318:...
262:552
259:165
218:19
68:or
48:or
40:In
1203::
829:(4
814:(2
799:(0
789:(7
779:(5
769:(3
759:(0
691:(6
676:(5
640:18
638:(8
628:(7
602:(6
592:(5
582:(4
499:.
374:26
256:49
253:21
215:18
212:17
209:16
206:15
203:14
200:13
197:12
194:11
191:10
838:)
834:1
823:)
819:1
808:)
804:1
793:)
791:1
783:)
781:1
773:)
771:1
763:)
761:1
700:)
696:2
685:)
681:1
652:)
642:)
632:)
630:4
620:3
618:6
612:2
610:6
606:)
604:1
596:)
594:2
586:)
584:1
565:)
557:(
547:e
540:t
533:v
520:.
510:"
505:.
371:8
368:5
365:2
362:1
359:1
356:0
353:0
315:5
312:1
309:2
306:0
303:0
300:0
297:0
294:0
250:7
247:3
244:2
241:1
238:1
235:0
232:0
227:n
188:9
185:8
182:7
179:6
176:5
173:4
170:3
167:2
164:1
160:n
126:n
122:n
96:q
92:p
88:q
84:p
20:)
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