116:. Later knot tables took two approaches to resolving this: some just skipped one of the entries without renumbering, and others renumbered the later entries to remove the hole. The resulting ambiguity has continued to the present day, and has been further compounded by mistaken attempts to correct errors caused by this that were themselves incorrect. For example, Wolfram Web's Perko Pair page erroneously compares two different knots (due to the renumbering by mathematicians such as Burde and Bar-Natan).
1041:
1053:
20:
132:
used computer searches to count all knots with 16 or fewer crossings. This research was performed separately using two different algorithms on different computers, lending support to the correctness of its results. Both counts found 1701936
59:
have been tabulated. The major challenge of the process is that many apparently different knots may actually be different geometrical presentations of the same topological entity, and that proving or disproving
249:
344:. Leibniz International Proceedings in Informatics (LIPIcs). Vol. 164. Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum für Informatik. pp. 25:1–25:17.
98:
and others started to attempt to count all possible knots. Because their work predated the invention of the digital computer, all work had to be done by hand.
141:) with up to 16 crossings. Most recently, in 2020, Benjamin Burton classified all prime knots up to 19 crossings (of which there are almost 300 million).
313:
155:
144:
Starting with three crossings (the minimum for any nontrivial knot), the number of prime knots for each number of crossings is
418:
359:
210:
986:
905:
267:
86:
made a hypothesis that the chemical elements were based upon knotted vortices in the aether. In an attempt to make a
83:
452:
87:
900:
895:
771:
125:
56:
32:
1057:
472:
304:
376:
534:
604:
599:
540:
411:
77:
732:
946:
915:
73:
44:
776:
129:
1079:
1045:
816:
404:
237:
8:
853:
836:
181:
40:
874:
821:
435:
431:
241:
186:
176:
48:
337:
112:
In 1974, Kenneth Perko discovered a duplication in the Tait-Little tables, called the
971:
920:
870:
826:
786:
781:
699:
355:
245:
1006:
831:
727:
462:
345:
283:
225:
148:
1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705, ... (sequence
61:
162:
Modern automated methods can now enumerate billions of knots in a matter of days.
966:
865:
811:
766:
759:
649:
561:
444:
233:
396:
350:
1026:
925:
887:
806:
719:
594:
586:
546:
287:
1073:
961:
749:
742:
737:
976:
956:
860:
843:
639:
576:
95:
991:
754:
659:
528:
508:
498:
490:
482:
427:
171:
28:
1011:
996:
951:
848:
801:
796:
791:
621:
518:
229:
134:
113:
107:
91:
52:
24:
1016:
684:
271:
1001:
611:
342:
36th
International Symposium on Computational Geometry (SoCG 2020)
47:, mathematicians have tried to classify and tabulate all possible
1021:
669:
629:
138:
910:
19:
981:
150:
209:
Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998),
208:
306:
16:
Attempt to classify and tabulate all possible knots
426:
1071:
64:is much more difficult than it at first seems.
412:
340:. In Cabello, Sergio; Chen, Danny Z. (eds.).
276:Proceedings of the Royal Society of Edinburgh
419:
405:
349:
374:
18:
298:
296:
266:
1072:
335:
400:
204:
202:
1052:
293:
13:
199:
14:
1091:
375:Richeson, David S. (2022-10-31).
302:
1051:
1040:
1039:
377:"Why Mathematicians Study Knots"
319:from the original on 2019-05-30
255:from the original on 2010-07-29
906:Dowker–Thistlethwaite notation
368:
329:
260:
218:The Mathematical Intelligencer
119:
88:periodic table of the elements
1:
192:
101:
67:
338:"The Next 350 Million Knots"
336:Burton, Benjamin A. (2020).
7:
351:10.4230/LIPIcs.SoCG.2020.25
211:"The first 1,701,936 knots"
165:
10:
1096:
105:
71:
1035:
939:
896:Alexander–Briggs notation
883:
718:
620:
585:
443:
288:10.1017/s0370164600045430
82:In the 19th century, Sir
78:Vortex theory of the atom
987:List of knots and links
535:Kinoshita–Terasaka knot
74:History of knot theory
51:. As of May 2008, all
36:
777:Finite type invariant
130:Morwen Thistlethwaite
23:A small table of all
22:
947:Alexander's theorem
182:List of prime knots
41:Sir William Thomson
230:10.1007/BF03025227
187:Unknotting problem
177:Knot (mathematics)
37:
1067:
1066:
921:Reidemeister move
787:Khovanov homology
782:Hyperbolic volume
361:978-3-95977-143-6
272:"On vortex atoms"
1087:
1055:
1054:
1043:
1042:
1007:Tait conjectures
710:
709:
695:
694:
680:
679:
572:
571:
557:
556:
541:(−2,3,7) pretzel
421:
414:
407:
398:
397:
391:
390:
388:
387:
372:
366:
365:
353:
333:
327:
326:
325:
324:
318:
311:
300:
291:
290:
268:Thomson, William
264:
258:
256:
254:
215:
206:
153:
62:knot equivalence
1095:
1094:
1090:
1089:
1088:
1086:
1085:
1084:
1070:
1069:
1068:
1063:
1031:
935:
901:Conway notation
885:
879:
866:Tricolorability
714:
708:
705:
704:
703:
693:
690:
689:
688:
678:
675:
674:
673:
665:
655:
645:
635:
616:
595:Composite knots
581:
570:
567:
566:
565:
562:Borromean rings
555:
552:
551:
550:
524:
514:
504:
494:
486:
478:
468:
458:
439:
425:
395:
394:
385:
383:
381:Quanta Magazine
373:
369:
362:
334:
330:
322:
320:
316:
309:
301:
294:
265:
261:
252:
213:
207:
200:
195:
168:
149:
137:(including the
122:
110:
104:
84:William Thomson
80:
72:Main articles:
70:
17:
12:
11:
5:
1093:
1083:
1082:
1065:
1064:
1062:
1061:
1049:
1036:
1033:
1032:
1030:
1029:
1027:Surgery theory
1024:
1019:
1014:
1009:
1004:
999:
994:
989:
984:
979:
974:
969:
964:
959:
954:
949:
943:
941:
937:
936:
934:
933:
928:
926:Skein relation
923:
918:
913:
908:
903:
898:
892:
890:
881:
880:
878:
877:
871:Unknotting no.
868:
863:
858:
857:
856:
846:
841:
840:
839:
834:
829:
824:
819:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
763:
762:
752:
747:
746:
745:
735:
730:
724:
722:
716:
715:
713:
712:
706:
697:
691:
682:
676:
667:
663:
657:
653:
647:
643:
637:
633:
626:
624:
618:
617:
615:
614:
609:
608:
607:
602:
591:
589:
583:
582:
580:
579:
574:
568:
559:
553:
544:
538:
532:
526:
522:
516:
512:
506:
502:
496:
492:
488:
484:
480:
476:
470:
466:
460:
456:
449:
447:
441:
440:
424:
423:
416:
409:
401:
393:
392:
367:
360:
328:
292:
259:
197:
196:
194:
191:
190:
189:
184:
179:
174:
167:
164:
160:
159:
121:
118:
106:Main article:
103:
100:
69:
66:
15:
9:
6:
4:
3:
2:
1092:
1081:
1078:
1077:
1075:
1060:
1059:
1050:
1048:
1047:
1038:
1037:
1034:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
963:
962:Conway sphere
960:
958:
955:
953:
950:
948:
945:
944:
942:
938:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
893:
891:
889:
882:
876:
872:
869:
867:
864:
862:
859:
855:
852:
851:
850:
847:
845:
842:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
814:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
761:
758:
757:
756:
753:
751:
748:
744:
741:
740:
739:
736:
734:
733:Arf invariant
731:
729:
726:
725:
723:
721:
717:
701:
698:
686:
683:
671:
668:
661:
658:
651:
648:
641:
638:
631:
628:
627:
625:
623:
619:
613:
610:
606:
603:
601:
598:
597:
596:
593:
592:
590:
588:
584:
578:
575:
563:
560:
548:
545:
542:
539:
536:
533:
530:
527:
520:
517:
510:
507:
500:
497:
495:
489:
487:
481:
474:
471:
464:
461:
454:
451:
450:
448:
446:
442:
437:
433:
429:
422:
417:
415:
410:
408:
403:
402:
399:
382:
378:
371:
363:
357:
352:
347:
343:
339:
332:
315:
308:
307:
299:
297:
289:
285:
281:
277:
273:
269:
263:
251:
247:
243:
239:
235:
231:
227:
223:
219:
212:
205:
203:
198:
188:
185:
183:
180:
178:
175:
173:
170:
169:
163:
157:
152:
147:
146:
145:
142:
140:
136:
131:
127:
117:
115:
109:
99:
97:
93:
89:
85:
79:
75:
65:
63:
58:
54:
50:
46:
45:vortex theory
42:
34:
30:
29:mirror images
26:
21:
1056:
1044:
972:Double torus
957:Braid theory
930:
772:Crossing no.
767:Crosscap no.
453:Figure-eight
384:. Retrieved
380:
370:
341:
331:
321:, retrieved
305:
303:Hoste, Jim,
279:
275:
262:
224:(4): 33–48,
221:
217:
161:
143:
123:
111:
96:C. N. Little
81:
38:
1080:Knot theory
807:Linking no.
728:Alternating
529:Conway knot
509:Carrick mat
463:Three-twist
428:Knot theory
172:Knot theory
135:prime knots
124:Jim Hoste,
120:New methods
53:prime knots
39:Ever since
27:(excluding
25:prime knots
967:Complement
931:Tabulation
888:operations
812:Polynomial
802:Link group
797:Knot group
760:Invertible
738:Bridge no.
720:Invariants
650:Cinquefoil
519:Perko pair
445:Hyperbolic
386:2022-11-05
323:2020-06-27
282:: 94–105,
193:References
126:Jeff Weeks
114:Perko pair
108:Perko pair
102:Perko pair
92:P. G. Tait
68:Beginnings
861:Stick no.
817:Alexander
755:Chirality
700:Solomon's
660:Septafoil
587:Satellite
547:Whitehead
473:Stevedore
57:crossings
55:up to 16
35:or fewer.
33:crossings
31:) with 7
1074:Category
1046:Category
916:Mutation
884:Notation
837:Kauffman
750:Brunnian
743:2-bridge
612:Knot sum
543:(12n242)
314:archived
270:(1869),
250:archived
246:18027155
166:See also
1058:Commons
977:Fibered
875:problem
844:Pretzel
822:Bracket
640:Trefoil
577:L10a140
537:(11n42)
531:(11n34)
499:Endless
238:1646740
154:in the
151:A002863
1022:Writhe
992:Ribbon
827:HOMFLY
670:Unlink
630:Unknot
605:Square
600:Granny
358:
244:
236:
139:unknot
128:, and
1012:Twist
997:Slice
952:Berge
940:Other
911:Flype
849:Prime
832:Jones
792:Genus
622:Torus
436:links
432:knots
317:(PDF)
310:(PDF)
253:(PDF)
242:S2CID
214:(PDF)
49:knots
1017:Wild
982:Knot
886:and
873:and
854:list
685:Hopf
434:and
356:ISBN
156:OEIS
76:and
1002:Sum
523:161
521:(10
346:doi
284:doi
226:doi
43:'s
1076::
702:(4
687:(2
672:(0
662:(7
652:(5
642:(3
632:(0
564:(6
549:(5
513:18
511:(8
501:(7
475:(6
465:(5
455:(4
379:.
354:.
312:,
295:^
278:,
274:,
248:,
240:,
234:MR
232:,
222:20
220:,
216:,
201:^
94:,
90:,
711:)
707:1
696:)
692:1
681:)
677:1
666:)
664:1
656:)
654:1
646:)
644:1
636:)
634:1
573:)
569:2
558:)
554:1
525:)
515:)
505:)
503:4
493:3
491:6
485:2
483:6
479:)
477:1
469:)
467:2
459:)
457:1
438:)
430:(
420:e
413:t
406:v
389:.
364:.
348::
286::
280:6
257:.
228::
158:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
