25:. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an
303:
1121:
49:); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
1013:
979:
174:
1111:
879:
802:
214:; a knot that can be represented by an alternating diagram (i.e. the crossing alternate over and under as one traverses the knot).
809:
620:
876:
91:
816:
1090:
1085:
1080:
1075:
1070:
1065:
1060:
1055:
1050:
1045:
1040:
1033:
1022:
925:
788:
693:
169:
1006:
152:, a compact 3 manifold obtained by removing an open neighborhood of a proper embedding of a tame knot from the 3-sphere.
116:
799:
920:
194:
806:
795:
601:
813:
609:
is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.
101:
97:
785:
483:
698:
628:
is a polynomial invariant of framed links. Related to the Jones polynomial. Also known as the
Kauffman bracket.
488:
844:
999:
957:
872:
851:
678:
673:
449:
967:
962:
895:
823:
58:
473:
858:
792:
781:
598:
493:
41:. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of
578:
279:
1116:
900:
837:
703:
631:
612:
567:
463:
441:
458:
984:
742:
737:
665:
645:
593:
237:
573:
940:
869:
848:
841:
718:
655:
625:
352:
69:
22:
468:
1126:
945:
935:
820:
713:
641:
555:
387:
306:
247:
159:
915:
910:
890:
855:
778:
651:
616:
446:
211:
155:
126:
930:
886:
865:
834:
830:
732:
688:
606:
543:
478:
414:
369:
337:
273:
263:
149:
85:
46:
34:
595:
is an invariant defined on knots which is invariant under ambient isotopies of the knot.
905:
747:
708:
635:
590:
534:
453:
409:
393:
322:
312:
234:
1105:
757:
683:
375:
206:
188:
227:
and defined in terms of their properties with respect to a genus 2 Heegaard surface.
505:
419:
347:
253:
224:
136:
105:
80:
603:
is a knot invariant that can be extended to an invariant of certain singular knots
752:
510:
429:
317:
258:
241:
17:
883:
862:
827:
761:
727:
424:
342:
332:
327:
268:
230:
221:
217:
184:
120:
110:
76:
722:
381:
357:
26:
550:
524:
760:, the number of arcs that must be added to make the knot complement a
991:
766:
399:
142:
30:
250:, a knot that can be embedded in a double torus (a genus 2 surface).
113:, two entries in a knot table that were later shown to be identical.
529:
129:
is the twist knot with three-half twists, also known as the 5
995:
378:, a set of links which become trivial if one loop is removed
619:; The first knot polynomial (1923). Sometimes called the
72:
gives a general introduction to the concept of a knot.
668:
is a 2-variable knot polynomial due to Louis
Kauffman.
282:
244:
is knot which is not equivalent to its mirror image.
297:
396:, a twisted loop linked with an untwisted loop.
158:general introduction to knots with mention of
1007:
8:
562:Elementary treatment using polygonal curves
1014:
1000:
992:
289:
285:
284:
281:
88:also known as a (5, 2) torus knot.
390:, a two-ring link with four crossings.
7:
980:List of mathematical knots and links
139:A knot with crossing number 3
14:
1122:Outlines of mathematics and logic
298:{\displaystyle \mathbb {R} ^{3}}
1091:Technology and applied sciences
384:, the simplest non-trivial link
195:continued fraction regular form
92:Figure-eight knot (mathematics)
175:Dowker–Thistlethwaite notation
165:Notation used in knot theory:
1:
1066:Natural and physical sciences
800:Eilenberg–Mazur swindle
877:Milnor conjecture (topology)
372:, the simplest Brunnian link
220:a class of knots related to
117:Stevedore knot (mathematics)
1086:Society and social sciences
1081:Religion and belief systems
621:Alexander–Conway polynomial
570:(R1 move, R2 move, R3 move)
123:with crossing number 6
104:are a connected sum of two
1143:
786:Birman–Wenzl algebra
484:Tangle denominator closure
56:
1112:Mathematics-related lists
1029:
926:Thurston–Bennequin number
585:Invariants and properties
102:Square knot (mathematics)
98:Granny knot (mathematics)
45:upon itself (known as an
1023:Knowledge (XXG) outlines
489:Tangle numerator closure
405:General types of links:
305:tangent to the standard
94:the only 4-crossing knot
1076:Philosophy and thinking
679:Average crossing number
674:Arf invariant of a knot
921:Temperley–Lieb algebra
896:Mutation (knot theory)
299:
276:are knots embedded in
75:Two classes of knots:
59:History of knot theory
1061:Mathematics and logic
814:Gordon–Luecke theorem
793:Clasper (mathematics)
773:Mathematical problems
599:Finite type invariant
300:
1046:Geography and places
1041:Culture and the arts
901:Physical knot theory
704:Kontsevich invariant
662:to the knot or link.
613:Alexander polynomial
442:Tangle (mathematics)
280:
64:Knots, links, braids
985:List of prime knots
968:Alexander's theorem
807:Fáry–Milnor theorem
743:Signature of a knot
738:Self-linking number
666:Kauffman polynomial
646:HOMFLYPT polynomial
615:and the associated
474:Inverse of a tangle
1056:History and events
1051:Health and fitness
958:Schubert's theorem
941:Unknotting problem
870:Link (knot theory)
849:Linkless embedding
842:Knotless embedding
719:Racks and quandles
656:Laurent polynomial
626:Bracket polynomial
295:
201:General knot types
160:Reidemeister moves
70:Knot (mathematics)
23:mathematical knots
1099:
1098:
1034:General reference
946:Volume conjecture
936:Unknotting number
821:Khovanov homology
714:Milnor invariants
699:Hyperbolic volume
642:Homfly polynomial
632:Conway polynomial
556:Reidemeister move
494:Reciprocal tangle
307:contact structure
248:Double torus knot
33:in 3-dimensional
1134:
1016:
1009:
1002:
993:
963:Conway's theorem
916:Tait conjectures
911:Smith conjecture
856:Link concordance
779:Berge conjecture
658:in the variable
652:Jones polynomial
617:Alexander matrix
579:delta-equivalent
447:Algebraic tangle
304:
302:
301:
296:
294:
293:
288:
212:Alternating knot
156:Knots and graphs
127:Three-twist knot
21:is the study of
1142:
1141:
1137:
1136:
1135:
1133:
1132:
1131:
1102:
1101:
1100:
1095:
1071:People and self
1025:
1020:
976:
954:
931:Tricolorability
835:Knot tabulation
775:
733:Seifert surface
694:Crossing number
689:Crosscap number
636:Skein relations
607:Knot polynomial
587:
568:elementary move
564:
544:Tricolorability
521:
502:
479:Rational tangle
464:Tangle rotation
438:
415:Hyperbolic link
370:Borromean rings
366:
338:Transverse knot
283:
278:
277:
274:Legendrian knot
264:Invertible knot
203:
170:Conway notation
150:Knot complement
132:
86:Cinquefoil knot
66:
61:
55:
47:ambient isotopy
35:Euclidean space
12:
11:
5:
1140:
1138:
1130:
1129:
1124:
1119:
1114:
1104:
1103:
1097:
1096:
1094:
1093:
1088:
1083:
1078:
1073:
1068:
1063:
1058:
1053:
1048:
1043:
1037:
1036:
1030:
1027:
1026:
1021:
1019:
1018:
1011:
1004:
996:
990:
989:
988:
987:
975:
972:
971:
970:
965:
960:
953:
950:
949:
948:
943:
938:
933:
928:
923:
918:
913:
908:
906:Planar algebra
903:
898:
893:
888:
881:
874:
867:
860:
853:
846:
839:
832:
825:
818:
811:
804:
797:
790:
783:
774:
771:
770:
769:
764:
755:
750:
748:Skein relation
745:
740:
735:
730:
725:
716:
711:
709:Linking number
706:
701:
696:
691:
686:
681:
676:
671:
670:
669:
663:
649:
639:
629:
623:
604:
596:
591:Knot invariant
586:
583:
582:
581:
576:
571:
563:
560:
559:
558:
553:
548:
547:
546:
532:
527:
520:
517:
516:
515:
514:
513:
501:
498:
497:
496:
491:
486:
481:
476:
471:
466:
461:
459:Tangle product
456:
454:Tangle diagram
451:
444:
437:
434:
433:
432:
427:
422:
417:
412:
410:Algebraic link
403:
402:
397:
394:Whitehead link
391:
388:Solomon's knot
385:
379:
373:
365:
362:
361:
360:
355:
350:
345:
340:
335:
330:
325:
323:Satellite knot
320:
315:
313:Lissajous knot
310:
292:
287:
271:
266:
261:
256:
251:
245:
239:
235:Satellite knot
228:
215:
209:
202:
199:
198:
197:
192:
189:Gauss diagrams
182:
172:
163:
162:
153:
147:
146:
145:
140:
134:
130:
124:
114:
108:
95:
89:
83:
65:
62:
57:Main article:
54:
51:
13:
10:
9:
6:
4:
3:
2:
1139:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1109:
1107:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1052:
1049:
1047:
1044:
1042:
1039:
1038:
1035:
1032:
1031:
1028:
1024:
1017:
1012:
1010:
1005:
1003:
998:
997:
994:
986:
983:
982:
981:
978:
977:
973:
969:
966:
964:
961:
959:
956:
955:
951:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
892:
891:Möbius energy
889:
887:
885:
882:
880:
878:
875:
873:
871:
868:
866:
864:
861:
859:
857:
854:
852:
850:
847:
845:
843:
840:
838:
836:
833:
831:
829:
826:
824:
822:
819:
817:
815:
812:
810:
808:
805:
803:
801:
798:
796:
794:
791:
789:
787:
784:
782:
780:
777:
776:
772:
768:
765:
763:
759:
758:Tunnel number
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
720:
717:
715:
712:
710:
707:
705:
702:
700:
697:
695:
692:
690:
687:
685:
684:Bridge number
682:
680:
677:
675:
672:
667:
664:
661:
657:
653:
650:
647:
643:
640:
637:
633:
630:
627:
624:
622:
618:
614:
611:
610:
608:
605:
602:
600:
597:
594:
592:
589:
588:
584:
580:
577:
575:
572:
569:
566:
565:
561:
557:
554:
552:
549:
545:
542:
541:
540:
538:
533:
531:
528:
526:
523:
522:
518:
512:
509:
508:
507:
504:
503:
499:
495:
492:
490:
487:
485:
482:
480:
477:
475:
472:
470:
467:
465:
462:
460:
457:
455:
452:
450:
448:
445:
443:
440:
439:
435:
431:
428:
426:
423:
421:
418:
416:
413:
411:
408:
407:
406:
401:
398:
395:
392:
389:
386:
383:
380:
377:
376:Brunnian link
374:
371:
368:
367:
363:
359:
356:
354:
351:
349:
346:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
314:
311:
308:
290:
275:
272:
270:
267:
265:
262:
260:
257:
255:
252:
249:
246:
243:
240:
238:
236:
232:
229:
226:
223:
219:
216:
213:
210:
208:
207:2-bridge knot
205:
204:
200:
196:
193:
190:
186:
183:
180:
176:
173:
171:
168:
167:
166:
161:
157:
154:
151:
148:
144:
141:
138:
135:
128:
125:
122:
118:
115:
112:
109:
107:
106:Trefoil knots
103:
99:
96:
93:
90:
87:
84:
82:
81:pretzel knots
78:
74:
73:
71:
68:
67:
63:
60:
52:
50:
48:
44:
40:
36:
32:
28:
24:
20:
19:
659:
574:R-equivalent
536:
506:Braid theory
420:Pretzel link
404:
348:Virtual knot
254:Fibered knot
178:
164:
137:Trefoil knot
42:
38:
16:
15:
1117:Knot theory
753:Slice genus
511:Braid group
430:String link
353:welded knot
318:Ribbon knot
259:Framed knot
242:Chiral knot
179:DT notation
133: knot.
77:torus knots
18:Knot theory
1106:Categories
884:Milnor map
863:Link group
828:Knot group
762:handlebody
728:Ropelength
654:assigns a
519:Operations
469:Tangle sum
425:Split link
343:Twist knot
333:Torus knot
328:Slice knot
269:Prime knot
231:Cable knot
222:Lens space
218:Berge knot
187:(see also
185:Gauss code
121:prime knot
111:Perko pair
723:Biquandle
539:-coloring
382:Hopf link
358:Wild knot
225:surgeries
27:embedding
1127:Outlines
952:Theorems
551:Knot sum
525:Band sum
436:Tangles
53:History
767:Writhe
500:Braids
400:Unlink
233:, see
143:Unknot
31:circle
974:Lists
634:uses
530:Flype
364:Links
29:of a
721:and
535:Fox
119:, a
100:and
79:and
644:or
1108::
37:,
1015:e
1008:t
1001:v
660:t
648:.
638:.
537:n
309:.
291:3
286:R
191:)
181:)
177:(
131:2
43:R
39:R
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