29:
1371:
136:
1383:
182:
The granny knot can be constructed from two identical trefoil knots, which must either be both left-handed or both right-handed. Each of the two knots is cut, and then the loose ends are joined together pairwise. The resulting connected sum is the granny knot.
573:
167:, which can also be described as a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the granny knot and the square knot are the simplest of all composite knots.
675:
287:
365:
108:
380:
186:
It is important that the original trefoil knots be identical to each another. If mirror-image trefoil knots are used instead, the result is a square knot.
198:
of a granny knot is six, which is the smallest possible crossing number for a composite knot. Unlike the square knot, the granny knot is not a
578:
This is the square of the Jones polynomial for the right-handed trefoil knot, and is different from the Jones polynomial for a square knot.
591:
297:
1429:
748:
1414:
1316:
1424:
1419:
1235:
219:
782:
306:
1230:
1225:
1101:
195:
46:
1387:
802:
864:
934:
870:
741:
685:
164:
156:
66:
73:
1062:
1276:
1245:
1106:
568:{\displaystyle V(q)=(q^{-1}+q^{-3}-q^{-4})^{2}=q^{-2}+2q^{-4}-2q^{-5}+q^{-6}-2q^{-7}+q^{-8}.}
1409:
1375:
1146:
734:
210:
8:
1183:
1166:
1204:
1151:
765:
761:
1301:
1250:
1200:
1156:
1116:
1111:
1029:
706:
681:
127:
1336:
1161:
1057:
792:
371:
293:
119:
688:, and is the simplest example of two different knots with isomorphic knot groups.
1296:
1260:
1195:
1141:
1096:
1089:
979:
891:
774:
726:
709:
1356:
1255:
1217:
1136:
1049:
924:
916:
876:
370:
These two polynomials are the same as those for the square knot. However, the
152:
123:
28:
1403:
1291:
1079:
1072:
1067:
1306:
1286:
1190:
1173:
969:
906:
160:
56:
1321:
1084:
989:
858:
838:
828:
820:
812:
757:
199:
171:
144:
39:
1341:
1326:
1281:
1178:
1131:
1126:
1121:
951:
848:
582:
203:
1346:
1014:
714:
1331:
941:
135:
1351:
999:
959:
296:
of the
Alexander polynomial of a trefoil knot. Similarly, the
1240:
1311:
670:{\displaystyle \langle x,y,z\mid xyx=yxy,xzx=zxz\rangle .}
170:
The granny knot is the mathematical version of the common
16:
Connected sum of two trefoil knots with same chirality
594:
383:
309:
222:
76:
704:
669:
567:
359:
281:
102:
756:
1401:
585:of the granny knot is given by the presentation
742:
661:
595:
282:{\displaystyle \Delta (t)=(t-1+t^{-1})^{2},}
749:
735:
27:
360:{\displaystyle \nabla (z)=(z^{2}+1)^{2}.}
134:
1402:
374:for the (right-handed) granny knot is
730:
705:
1382:
13:
310:
223:
87:
14:
1441:
1381:
1370:
1369:
163:. It is closely related to the
177:
1236:Dowker–Thistlethwaite notation
698:
448:
399:
393:
387:
345:
325:
319:
313:
267:
238:
232:
226:
1:
691:
189:
1430:Tricolorable knots and links
103:{\displaystyle 3_{1}\#3_{1}}
7:
1415:Alternating knots and links
10:
1446:
1425:Unfibered knots and links
1420:Composite knots and links
1365:
1269:
1226:Alexander–Briggs notation
1213:
1048:
950:
915:
773:
684:to the knot group of the
118:
113:
65:
55:
45:
35:
26:
21:
1317:List of knots and links
865:Kinoshita–Terasaka knot
155:obtained by taking the
671:
569:
361:
283:
213:of the granny knot is
140:
104:
1107:Finite type invariant
672:
570:
362:
284:
138:
105:
592:
381:
307:
300:of a granny knot is
292:which is simply the
220:
211:Alexander polynomial
74:
1277:Alexander's theorem
707:Weisstein, Eric W.
667:
565:
357:
279:
141:
100:
1397:
1396:
1251:Reidemeister move
1117:Khovanov homology
1112:Hyperbolic volume
298:Conway polynomial
159:of two identical
133:
132:
1437:
1385:
1384:
1373:
1372:
1337:Tait conjectures
1040:
1039:
1025:
1024:
1010:
1009:
902:
901:
887:
886:
871:(−2,3,7) pretzel
751:
744:
737:
728:
727:
721:
720:
719:
702:
676:
674:
673:
668:
574:
572:
571:
566:
561:
560:
545:
544:
526:
525:
510:
509:
491:
490:
472:
471:
456:
455:
446:
445:
430:
429:
414:
413:
372:Jones polynomial
366:
364:
363:
358:
353:
352:
337:
336:
288:
286:
285:
280:
275:
274:
265:
264:
109:
107:
106:
101:
99:
98:
86:
85:
31:
19:
18:
1445:
1444:
1440:
1439:
1438:
1436:
1435:
1434:
1400:
1399:
1398:
1393:
1361:
1265:
1231:Conway notation
1215:
1209:
1196:Tricolorability
1044:
1038:
1035:
1034:
1033:
1023:
1020:
1019:
1018:
1008:
1005:
1004:
1003:
995:
985:
975:
965:
946:
925:Composite knots
911:
900:
897:
896:
895:
892:Borromean rings
885:
882:
881:
880:
854:
844:
834:
824:
816:
808:
798:
788:
769:
755:
725:
724:
703:
699:
694:
593:
590:
589:
553:
549:
537:
533:
518:
514:
502:
498:
483:
479:
464:
460:
451:
447:
438:
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422:
418:
406:
402:
382:
379:
378:
348:
344:
332:
328:
308:
305:
304:
270:
266:
257:
253:
221:
218:
217:
196:crossing number
192:
180:
94:
90:
81:
77:
75:
72:
71:
17:
12:
11:
5:
1443:
1433:
1432:
1427:
1422:
1417:
1412:
1395:
1394:
1392:
1391:
1379:
1366:
1363:
1362:
1360:
1359:
1357:Surgery theory
1354:
1349:
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1334:
1329:
1324:
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1314:
1309:
1304:
1299:
1294:
1289:
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1273:
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1264:
1263:
1258:
1256:Skein relation
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1248:
1243:
1238:
1233:
1228:
1222:
1220:
1211:
1210:
1208:
1207:
1201:Unknotting no.
1198:
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1154:
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1114:
1109:
1104:
1099:
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1077:
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1065:
1060:
1054:
1052:
1046:
1045:
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1042:
1036:
1027:
1021:
1012:
1006:
997:
993:
987:
983:
977:
973:
967:
963:
956:
954:
948:
947:
945:
944:
939:
938:
937:
932:
921:
919:
913:
912:
910:
909:
904:
898:
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883:
874:
868:
862:
856:
852:
846:
842:
836:
832:
826:
822:
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814:
810:
806:
800:
796:
790:
786:
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771:
770:
754:
753:
746:
739:
731:
723:
722:
696:
695:
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690:
678:
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666:
663:
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654:
651:
648:
645:
642:
639:
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633:
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627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
576:
575:
564:
559:
556:
552:
548:
543:
540:
536:
532:
529:
524:
521:
517:
513:
508:
505:
501:
497:
494:
489:
486:
482:
478:
475:
470:
467:
463:
459:
454:
450:
444:
441:
437:
433:
428:
425:
421:
417:
412:
409:
405:
401:
398:
395:
392:
389:
386:
368:
367:
356:
351:
347:
343:
340:
335:
331:
327:
324:
321:
318:
315:
312:
290:
289:
278:
273:
269:
263:
260:
256:
252:
249:
246:
243:
240:
237:
234:
231:
228:
225:
191:
188:
179:
176:
153:composite knot
131:
130:
116:
115:
111:
110:
97:
93:
89:
84:
80:
69:
63:
62:
59:
53:
52:
49:
43:
42:
37:
33:
32:
24:
23:
15:
9:
6:
4:
3:
2:
1442:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1411:
1408:
1407:
1405:
1390:
1389:
1380:
1378:
1377:
1368:
1367:
1364:
1358:
1355:
1353:
1350:
1348:
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1310:
1308:
1305:
1303:
1300:
1298:
1295:
1293:
1292:Conway sphere
1290:
1288:
1285:
1283:
1280:
1278:
1275:
1274:
1272:
1268:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1224:
1223:
1221:
1219:
1212:
1206:
1202:
1199:
1197:
1194:
1192:
1189:
1185:
1182:
1181:
1180:
1177:
1175:
1172:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1144:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1091:
1088:
1087:
1086:
1083:
1081:
1078:
1074:
1071:
1070:
1069:
1066:
1064:
1063:Arf invariant
1061:
1059:
1056:
1055:
1053:
1051:
1047:
1031:
1028:
1016:
1013:
1001:
998:
991:
988:
981:
978:
971:
968:
961:
958:
957:
955:
953:
949:
943:
940:
936:
933:
931:
928:
927:
926:
923:
922:
920:
918:
914:
908:
905:
893:
890:
878:
875:
872:
869:
866:
863:
860:
857:
850:
847:
840:
837:
830:
827:
825:
819:
817:
811:
804:
801:
794:
791:
784:
781:
780:
778:
776:
772:
767:
763:
759:
752:
747:
745:
740:
738:
733:
732:
729:
717:
716:
711:
710:"Granny Knot"
708:
701:
697:
689:
687:
683:
664:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
588:
587:
586:
584:
579:
562:
557:
554:
550:
546:
541:
538:
534:
530:
527:
522:
519:
515:
511:
506:
503:
499:
495:
492:
487:
484:
480:
476:
473:
468:
465:
461:
457:
452:
442:
439:
435:
431:
426:
423:
419:
415:
410:
407:
403:
396:
390:
384:
377:
376:
375:
373:
354:
349:
341:
338:
333:
329:
322:
316:
303:
302:
301:
299:
295:
276:
271:
261:
258:
254:
250:
247:
244:
241:
235:
229:
216:
215:
214:
212:
207:
205:
201:
197:
187:
184:
175:
173:
168:
166:
162:
161:trefoil knots
158:
157:connected sum
154:
150:
146:
137:
129:
125:
121:
117:
112:
95:
91:
82:
78:
70:
68:
64:
60:
58:
54:
50:
48:
44:
41:
38:
34:
30:
25:
20:
1386:
1374:
1302:Double torus
1287:Braid theory
1102:Crossing no.
1097:Crosscap no.
929:
783:Figure-eight
713:
700:
679:
580:
577:
369:
291:
208:
193:
185:
181:
178:Construction
169:
148:
142:
139:3D depiction
128:tricolorable
67:A–B notation
47:Crossing no.
1410:Knot theory
1137:Linking no.
1058:Alternating
859:Conway knot
839:Carrick mat
793:Three-twist
758:Knot theory
686:square knot
200:ribbon knot
172:granny knot
165:square knot
149:granny knot
145:knot theory
120:alternating
40:Granny knot
36:Common name
22:Granny knot
1404:Categories
1297:Complement
1261:Tabulation
1218:operations
1142:Polynomial
1132:Link group
1127:Knot group
1090:Invertible
1068:Bridge no.
1050:Invariants
980:Cinquefoil
849:Perko pair
775:Hyperbolic
692:References
682:isomorphic
583:knot group
204:slice knot
190:Properties
1191:Stick no.
1147:Alexander
1085:Chirality
1030:Solomon's
990:Septafoil
917:Satellite
877:Whitehead
803:Stevedore
715:MathWorld
662:⟩
614:∣
596:⟨
555:−
539:−
528:−
520:−
504:−
493:−
485:−
466:−
440:−
432:−
424:−
408:−
311:∇
259:−
245:−
224:Δ
124:composite
88:#
57:Stick no.
1376:Category
1246:Mutation
1214:Notation
1167:Kauffman
1080:Brunnian
1073:2-bridge
942:Knot sum
873:(12n242)
680:This is
1388:Commons
1307:Fibered
1205:problem
1174:Pretzel
1152:Bracket
970:Trefoil
907:L10a140
867:(11n42)
861:(11n34)
829:Endless
1352:Writhe
1322:Ribbon
1157:HOMFLY
1000:Unlink
960:Unknot
935:Square
930:Granny
294:square
147:, the
126:,
122:,
1342:Twist
1327:Slice
1282:Berge
1270:Other
1241:Flype
1179:Prime
1162:Jones
1122:Genus
952:Torus
766:links
762:knots
202:or a
151:is a
114:Other
1347:Wild
1312:Knot
1216:and
1203:and
1184:list
1015:Hopf
764:and
581:The
209:The
194:The
1332:Sum
853:161
851:(10
143:In
1406::
1032:(4
1017:(2
1002:(0
992:(7
982:(5
972:(3
962:(0
894:(6
879:(5
843:18
841:(8
831:(7
805:(6
795:(5
785:(4
712:.
206:.
174:.
1041:)
1037:1
1026:)
1022:1
1011:)
1007:1
996:)
994:1
986:)
984:1
976:)
974:1
966:)
964:1
903:)
899:2
888:)
884:1
855:)
845:)
835:)
833:4
823:3
821:6
815:2
813:6
809:)
807:1
799:)
797:2
789:)
787:1
768:)
760:(
750:e
743:t
736:v
718:.
665:.
659:z
656:x
653:z
650:=
647:x
644:z
641:x
638:,
635:y
632:x
629:y
626:=
623:x
620:y
617:x
611:z
608:,
605:y
602:,
599:x
563:.
558:8
551:q
547:+
542:7
535:q
531:2
523:6
516:q
512:+
507:5
500:q
496:2
488:4
481:q
477:2
474:+
469:2
462:q
458:=
453:2
449:)
443:4
436:q
427:3
420:q
416:+
411:1
404:q
400:(
397:=
394:)
391:q
388:(
385:V
355:.
350:2
346:)
342:1
339:+
334:2
330:z
326:(
323:=
320:)
317:z
314:(
277:,
272:2
268:)
262:1
255:t
251:+
248:1
242:t
239:(
236:=
233:)
230:t
227:(
96:1
92:3
83:1
79:3
61:8
51:6
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