584:
1757:
572:
685:
1769:
754:
761:
If the self-homeomorphism, α, reverses the orientation of the knot, it is said to be negative amphicheiral. This is equivalent to the knot being isotopic to the reverse of its mirror image. The noninvertible knot with this property that has the fewest crossings is the knot
744:
If the self-homeomorphism, α, preserves the orientation of the knot, it is said to be positive amphicheiral. This is equivalent to the knot being isotopic to its mirror. No knots with crossing number smaller than twelve are positive amphicheiral and noninvertible .
884:
76:
There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, invertible, positively amphicheiral noninvertible, negatively amphicheiral noninvertible, and fully amphicheiral invertible.
923:
109:
found all 12-crossing and many 14-crossing amphicheiral knots in the late 1910s. But a counterexample to Tait's conjecture, a 15-crossing amphicheiral knot, was found by
787:
1134:
1702:
1621:
1795:
1168:
733:
689:
1611:
1616:
1487:
826:
713:
134:
118:
102:
1188:
1250:
583:
53:
to its mirror image (when identical while reversed). An oriented knot that is equivalent to its mirror image is an
1256:
571:
1320:
1315:
1127:
732:
to both its reverse and its mirror image, it is fully amphicheiral. The simplest knot with this property is the
1800:
1448:
62:
1773:
1662:
1631:
845:
1492:
114:
106:
656:
A chiral knot that can be smoothly deformed to itself with the opposite orientation is classified as a
1761:
1532:
1120:
1056:
857:
607:
101:
found all amphicheiral knots up to 10 crossings and conjectured that all amphicheiral knots had even
1569:
1552:
1590:
1537:
1151:
1147:
1072:
1046:
982:
943:
904:
818:
763:
43:
717:
673:
110:
1687:
1636:
1586:
1542:
1502:
1497:
1415:
1004:
641:
1722:
1547:
1443:
1178:
1064:
974:
935:
896:
865:
802:
709:
611:
126:
50:
814:
716:. The first amphicheiral knot with odd crossing number is a 15-crossing knot discovered by
1682:
1646:
1581:
1527:
1482:
1475:
1365:
1277:
1160:
810:
729:
669:
657:
70:
1060:
861:
1742:
1641:
1603:
1522:
1435:
1310:
1302:
1262:
1102:
962:
645:
86:
66:
869:
17:
1789:
1677:
1465:
1458:
1453:
1076:
1007:
947:
908:
701:
697:
986:
822:
1692:
1672:
1576:
1559:
1355:
1292:
595:
563:
90:
1375:
1214:
1206:
1198:
1097:
69:. A knot's chirality can be further classified depending on whether or not it is
1707:
1244:
1224:
1143:
1112:
35:
31:
1727:
1712:
1667:
1564:
1517:
1512:
1507:
1337:
1234:
1068:
939:
900:
603:
122:
98:
1732:
1400:
1012:
684:
1029:
Ramadevi, P.; Govindarajan, T.R.; Kaul, R.K. (1994). "Chirality of Knots 9
1717:
1327:
705:
599:
94:
885:"XI.—On Knots, with a Census of the Amphicheirals with Twelve Crossings"
644:
is even better at detecting chirality, but there is no known polynomial
1051:
978:
806:
753:
85:
The possible chirality of certain knots was suspected since 1847 when
1737:
1385:
1345:
1626:
640:), then the knot is chiral, however the converse is not true. The
1697:
672:
or its mirror image, it is a fully chiral knot, for example the
186:
1116:
553:
501:
449:
397:
343:
291:
610:
cannot distinguish a knot from its mirror image, but the
961:
Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998).
786:
Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998),
121:
in 1998. However, Tait's conjecture was proven true for
131:
1655:
1599:
1434:
1336:
1301:
1159:
133:Number of knots of each type of chirality for each
708:, α, fixing the knot set-wise. All amphicheiral
27:Knot that is not equivalent to its mirror image
1128:
8:
998:
996:
456:Negative Amphicheiral Noninvertible knots
1135:
1121:
1113:
404:Positive Amphicheiral Noninvertible knots
1050:
696:An amphicheiral knot is one which has an
660:knot. Examples include the trefoil knot.
752:
683:
1092:
1090:
1088:
1086:
778:
560:
757:The first negative amphicheiral knot.
7:
1768:
668:If a knot is not equivalent to its
93:was chiral, and this was proven by
692:is the simplest amphicheiral knot.
598:, which was shown to be chiral by
25:
648:that can fully detect chirality.
1767:
1756:
1755:
846:"Classical Roots of Knot Theory"
594:The simplest chiral knot is the
582:
570:
922:Haseman, Mary Gertrude (1920).
883:Haseman, Mary Gertrude (1918).
1622:Dowker–Thistlethwaite notation
795:The Mathematical Intelligencer
589:The right-handed trefoil knot.
1:
870:10.1016/S0960-0779(97)00107-0
577:The left-handed trefoil knot.
1098:Three Dimensional Invariants
850:Chaos, Solitons and Fractals
844:Przytycki, Józef H. (1998).
1037:and Chern-Simons Theory"".
963:"The First 1,701,936 Knots"
924:"XXIII.—Amphicheiral Knots"
788:"The first 1,701,936 knots"
506:
454:
402:
348:
296:
244:
198:
192:
139:
1817:
1751:
1612:Alexander–Briggs notation
1069:10.1142/S0217732394003026
940:10.1017/S0080456800004476
901:10.1017/S0080456800012102
508:Fully Amphicheiral knots
1703:List of knots and links
1251:Kinoshita–Terasaka knot
1796:Chiral knots and links
1019:Accessed: May 5, 2013.
758:
693:
614:can in some cases; if
18:Fully amphichiral knot
1493:Finite type invariant
756:
749:Negative amphicheiral
740:Positive amphicheiral
687:
115:Morwen Thistlethwaite
107:Mary Gertrude Haseman
928:Trans. R. Soc. Edinb
889:Trans. R. Soc. Edinb
608:Alexander polynomial
141:Number of crossings
1663:Alexander's theorem
1061:1994MPLA....9.3205R
862:1998CSF.....9..531P
298:Fully chiral knots
137:
1039:Mod. Phys. Lett. A
1008:"Amphichiral Knot"
1005:Weisstein, Eric W.
979:10.1007/BF03025227
807:10.1007/BF03025227
759:
724:Fully amphicheiral
694:
351:Amphicheiral knots
132:
89:asserted that the
1783:
1782:
1637:Reidemeister move
1503:Khovanov homology
1498:Hyperbolic volume
734:figure-eight knot
710:alternating knots
690:figure-eight knot
680:Amphicheiral knot
664:Fully chiral knot
642:HOMFLY polynomial
602:. All nontrivial
559:
558:
246:Invertible knots
127:alternating knots
57:, also called an
55:amphicheiral knot
16:(Redirected from
1808:
1771:
1770:
1759:
1758:
1723:Tait conjectures
1426:
1425:
1411:
1410:
1396:
1395:
1288:
1287:
1273:
1272:
1257:(−2,3,7) pretzel
1137:
1130:
1123:
1114:
1107:
1094:
1081:
1080:
1054:
1026:
1020:
1018:
1017:
1000:
991:
990:
958:
952:
951:
919:
913:
912:
880:
874:
873:
841:
835:
833:
831:
825:, archived from
792:
783:
700:-reversing self-
612:Jones polynomial
606:are chiral. The
586:
574:
138:
21:
1816:
1815:
1811:
1810:
1809:
1807:
1806:
1805:
1786:
1785:
1784:
1779:
1747:
1651:
1617:Conway notation
1601:
1595:
1582:Tricolorability
1430:
1424:
1421:
1420:
1419:
1409:
1406:
1405:
1404:
1394:
1391:
1390:
1389:
1381:
1371:
1361:
1351:
1332:
1311:Composite knots
1297:
1286:
1283:
1282:
1281:
1278:Borromean rings
1271:
1268:
1267:
1266:
1240:
1230:
1220:
1210:
1202:
1194:
1184:
1174:
1155:
1141:
1111:
1110:
1095:
1084:
1045:(34): 3205–18.
1036:
1032:
1028:
1027:
1023:
1003:
1002:
1001:
994:
960:
959:
955:
921:
920:
916:
882:
881:
877:
856:(4/5): 531–45.
843:
842:
838:
829:
790:
785:
784:
780:
775:
767:
751:
742:
726:
714:crossing number
682:
666:
654:
652:Invertible knot
635:
622:
590:
587:
578:
575:
135:crossing number
103:crossing number
83:
65:of a knot is a
28:
23:
22:
15:
12:
11:
5:
1814:
1812:
1804:
1803:
1801:Knot chirality
1798:
1788:
1787:
1781:
1780:
1778:
1777:
1765:
1752:
1749:
1748:
1746:
1745:
1743:Surgery theory
1740:
1735:
1730:
1725:
1720:
1715:
1710:
1705:
1700:
1695:
1690:
1685:
1680:
1675:
1670:
1665:
1659:
1657:
1653:
1652:
1650:
1649:
1644:
1642:Skein relation
1639:
1634:
1629:
1624:
1619:
1614:
1608:
1606:
1597:
1596:
1594:
1593:
1587:Unknotting no.
1584:
1579:
1574:
1573:
1572:
1562:
1557:
1556:
1555:
1550:
1545:
1540:
1535:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1480:
1479:
1478:
1468:
1463:
1462:
1461:
1451:
1446:
1440:
1438:
1432:
1431:
1429:
1428:
1422:
1413:
1407:
1398:
1392:
1383:
1379:
1373:
1369:
1363:
1359:
1353:
1349:
1342:
1340:
1334:
1333:
1331:
1330:
1325:
1324:
1323:
1318:
1307:
1305:
1299:
1298:
1296:
1295:
1290:
1284:
1275:
1269:
1260:
1254:
1248:
1242:
1238:
1232:
1228:
1222:
1218:
1212:
1208:
1204:
1200:
1196:
1192:
1186:
1182:
1176:
1172:
1165:
1163:
1157:
1156:
1142:
1140:
1139:
1132:
1125:
1117:
1109:
1108:
1103:The Knot Atlas
1082:
1052:hep-th/9401095
1034:
1030:
1021:
992:
953:
934:(3): 597–602.
914:
875:
836:
777:
776:
774:
771:
765:
750:
747:
741:
738:
725:
722:
681:
678:
665:
662:
653:
650:
646:knot invariant
631:
618:
592:
591:
588:
581:
579:
576:
569:
567:
562:Both possible
557:
556:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
505:
504:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
466:
463:
460:
457:
453:
452:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
401:
400:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
347:
346:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
295:
294:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
250:
247:
243:
242:
239:
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233:
230:
227:
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221:
218:
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212:
209:
206:
203:
200:
197:
191:
190:
184:
181:
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175:
172:
169:
166:
163:
160:
157:
154:
151:
148:
145:
142:
87:Johann Listing
82:
79:
67:knot invariant
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1813:
1802:
1799:
1797:
1794:
1793:
1791:
1776:
1775:
1766:
1764:
1763:
1754:
1753:
1750:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1678:Conway sphere
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1660:
1658:
1654:
1648:
1645:
1643:
1640:
1638:
1635:
1633:
1630:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1609:
1607:
1605:
1598:
1592:
1588:
1585:
1583:
1580:
1578:
1575:
1571:
1568:
1567:
1566:
1563:
1561:
1558:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1530:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1499:
1496:
1494:
1491:
1489:
1486:
1484:
1481:
1477:
1474:
1473:
1472:
1469:
1467:
1464:
1460:
1457:
1456:
1455:
1452:
1450:
1449:Arf invariant
1447:
1445:
1442:
1441:
1439:
1437:
1433:
1417:
1414:
1402:
1399:
1387:
1384:
1377:
1374:
1367:
1364:
1357:
1354:
1347:
1344:
1343:
1341:
1339:
1335:
1329:
1326:
1322:
1319:
1317:
1314:
1313:
1312:
1309:
1308:
1306:
1304:
1300:
1294:
1291:
1279:
1276:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1236:
1233:
1226:
1223:
1216:
1213:
1211:
1205:
1203:
1197:
1190:
1187:
1180:
1177:
1170:
1167:
1166:
1164:
1162:
1158:
1153:
1149:
1145:
1138:
1133:
1131:
1126:
1124:
1119:
1118:
1115:
1105:
1104:
1099:
1093:
1091:
1089:
1087:
1083:
1078:
1074:
1070:
1066:
1062:
1058:
1053:
1048:
1044:
1040:
1025:
1022:
1015:
1014:
1009:
1006:
999:
997:
993:
988:
984:
980:
976:
972:
968:
964:
957:
954:
949:
945:
941:
937:
933:
929:
925:
918:
915:
910:
906:
902:
898:
895:(1): 235–55.
894:
890:
886:
879:
876:
871:
867:
863:
859:
855:
851:
847:
840:
837:
832:on 2013-12-15
828:
824:
820:
816:
812:
808:
804:
800:
796:
789:
782:
779:
772:
770:
768:
755:
748:
746:
739:
737:
735:
731:
728:If a knot is
723:
721:
719:
715:
711:
707:
703:
702:homeomorphism
699:
691:
686:
679:
677:
675:
671:
663:
661:
659:
651:
649:
647:
643:
639:
634:
630:
626:
621:
617:
613:
609:
605:
601:
597:
585:
580:
573:
568:
565:
564:trefoil knots
561:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
507:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
352:
349:
345:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
196:
193:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
143:
140:
136:
130:
128:
124:
120:
116:
112:
108:
104:
100:
96:
92:
88:
80:
78:
74:
72:
68:
64:
60:
56:
52:
49:
45:
41:
37:
33:
19:
1772:
1760:
1688:Double torus
1673:Braid theory
1488:Crossing no.
1483:Crosscap no.
1470:
1169:Figure-eight
1101:
1042:
1038:
1024:
1011:
973:(4): 33–48.
970:
967:Math. Intell
966:
956:
931:
927:
917:
892:
888:
878:
853:
849:
839:
827:the original
801:(4): 33–48,
798:
794:
781:
760:
743:
727:
695:
667:
655:
637:
632:
628:
624:
619:
615:
596:trefoil knot
593:
350:
195:Chiral knots
194:
84:
75:
59:achiral knot
58:
54:
47:
39:
32:mathematical
29:
1523:Linking no.
1444:Alternating
1245:Conway knot
1225:Carrick mat
1179:Three-twist
1144:Knot theory
698:orientation
604:torus knots
40:chiral knot
36:knot theory
1790:Categories
1683:Complement
1647:Tabulation
1604:operations
1528:Polynomial
1518:Link group
1513:Knot group
1476:Invertible
1454:Bridge no.
1436:Invariants
1366:Cinquefoil
1235:Perko pair
1161:Hyperbolic
773:References
712:have even
658:invertible
119:Jeff Weeks
99:P. G. Tait
81:Background
71:invertible
51:equivalent
1577:Stick no.
1533:Alexander
1471:Chirality
1416:Solomon's
1376:Septafoil
1303:Satellite
1263:Whitehead
1189:Stevedore
1077:119143024
1013:MathWorld
948:124014620
909:123957148
674:9 32 knot
627:) ≠
189:sequence
111:Jim Hoste
97:in 1914.
63:chirality
34:field of
1762:Category
1632:Mutation
1600:Notation
1553:Kauffman
1466:Brunnian
1459:2-bridge
1328:Knot sum
1259:(12n242)
987:18027155
823:18027155
730:isotopic
706:3-sphere
600:Max Dehn
340:1308449
238:1387166
95:Max Dehn
46:that is
1774:Commons
1693:Fibered
1591:problem
1560:Pretzel
1538:Bracket
1356:Trefoil
1293:L10a140
1253:(11n42)
1247:(11n34)
1215:Endless
1057:Bibcode
858:Bibcode
815:1646740
720:et al.
704:of the
670:inverse
554:A052400
502:A051768
450:A051767
398:A052401
344:A051766
337:226580
292:A051769
235:253292
91:trefoil
30:In the
1738:Writhe
1708:Ribbon
1543:HOMFLY
1386:Unlink
1346:Unknot
1321:Square
1316:Granny
1075:
1033:and 10
985:
946:
907:
821:
813:
334:37885
288:78717
285:26712
232:46698
117:, and
61:. The
1728:Twist
1713:Slice
1668:Berge
1656:Other
1627:Flype
1565:Prime
1548:Jones
1508:Genus
1338:Torus
1152:links
1148:knots
1073:S2CID
1047:arXiv
983:S2CID
944:S2CID
905:S2CID
830:(PDF)
819:S2CID
791:(PDF)
718:Hoste
498:1361
394:1539
331:6919
328:1103
282:8813
279:3069
276:1015
229:9988
226:2118
123:prime
42:is a
1733:Wild
1698:Knot
1602:and
1589:and
1570:list
1401:Hopf
1150:and
688:The
550:113
492:227
388:274
325:187
273:365
270:125
241:N/A
223:552
220:152
187:OEIS
44:knot
38:, a
1718:Sum
1239:161
1237:(10
1100:",
1065:doi
975:doi
936:doi
897:doi
866:doi
803:doi
544:41
538:17
486:40
446:65
382:58
376:13
322:27
267:47
264:16
217:49
214:16
183:16
180:15
177:14
174:13
171:12
168:11
165:10
48:not
1792::
1418:(4
1403:(2
1388:(0
1378:(7
1368:(5
1358:(3
1348:(0
1280:(6
1265:(5
1229:18
1227:(8
1217:(7
1191:(6
1181:(5
1171:(4
1085:^
1071:.
1063:.
1055:.
1041:.
1035:71
1031:42
1010:.
995:^
981:.
971:20
969:.
965:.
942:.
932:52
930:.
926:.
903:.
893:52
891:.
887:.
864:.
852:.
848:.
817:,
811:MR
809:,
799:20
797:,
793:,
769:.
766:17
736:.
676:.
547:0
541:0
535:0
532:7
529:0
526:4
523:0
520:1
517:0
514:1
511:0
495:1
489:0
483:0
480:6
477:0
474:1
471:0
468:0
465:0
462:0
459:0
443:0
440:6
437:0
434:1
431:0
428:0
425:0
422:0
419:0
416:0
413:0
410:0
407:0
391:1
385:0
379:0
373:0
370:5
367:0
364:1
361:0
358:1
355:0
319:2
316:0
313:0
310:0
307:0
304:0
301:0
261:7
258:2
255:2
252:0
249:1
211:7
208:2
205:2
202:0
199:1
162:9
159:8
156:7
153:6
150:5
147:4
144:3
129:.
125:,
113:,
105:.
73:.
1427:)
1423:1
1412:)
1408:1
1397:)
1393:1
1382:)
1380:1
1372:)
1370:1
1362:)
1360:1
1352:)
1350:1
1289:)
1285:2
1274:)
1270:1
1241:)
1231:)
1221:)
1219:4
1209:3
1207:6
1201:2
1199:6
1195:)
1193:1
1185:)
1183:2
1175:)
1173:1
1154:)
1146:(
1136:e
1129:t
1122:v
1106:.
1096:"
1079:.
1067::
1059::
1049::
1043:9
1016:.
989:.
977::
950:.
938::
911:.
899::
872:.
868::
860::
854:9
834:.
805::
764:8
638:q
636:(
633:k
629:V
625:q
623:(
620:k
616:V
566:.
20:)
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