1844:
1856:
513:
355:
729:
280:
956:
586:
Now we can define the Arf invariant of a knot to be 0 if it is pass-equivalent to the unknot, or 1 if it is pass-equivalent to the trefoil. This definition is equivalent to the one above.
810:
347:
888:
1069:
1027:
508:{\displaystyle \operatorname {Arf} (K)=\sum \limits _{i=1}^{g}\operatorname {lk} \left(a_{i},a_{i}^{+}\right)\operatorname {lk} \left(b_{i},b_{i}^{+}\right){\pmod {2}}.}
115:
547:
985:
1105:
66:
has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an embedded circle representing an element of the homology group. The
1071:
is an odd integer, it has to be congruent to 1 modulo 8. Combined with
Murasugi's result, this shows that the Arf invariant of a slice knot vanishes.
615:
1117:
Murasugi, Kunio, The Arf
Invariant for Knot Types, Proceedings of the American Mathematical Society, Vol. 21, No. 1. (Apr., 1969), pp. 69–72
172:
1221:
1789:
1708:
1190:
1164:
1142:
1255:
1698:
1703:
1574:
893:
1275:
597:
showed that the Arf invariant can be obtained by taking the partition function of a signed planar graph associated to a
1337:
583:; these two knots are not pass-equivalent and additionally, the right- and left-handed trefoils are pass-equivalent.
744:
1343:
1407:
1402:
1214:
288:
1882:
858:
1032:
990:
1860:
1749:
1718:
130:
1579:
1848:
1619:
1207:
735:
1656:
1639:
1677:
1624:
1238:
1234:
81:
1774:
1723:
1673:
1629:
1589:
1584:
1502:
1186:
1160:
1138:
159:
1809:
1634:
1530:
1265:
525:
1769:
1733:
1668:
1614:
1569:
1562:
1452:
1364:
1247:
1182:
961:
122:
36:
1829:
1728:
1690:
1609:
1522:
1397:
1389:
1349:
1152:
1130:
562:
118:
44:
32:
1876:
1764:
1552:
1545:
1540:
594:
67:
1779:
1759:
1663:
1646:
1442:
1379:
1174:
598:
580:
1462:
1301:
1293:
1285:
852:
From the Fox-Milnor criterion, which tells us that the
Alexander polynomial of a
1794:
1557:
1331:
1311:
1230:
1199:
519:
349:, is a symplectic basis for the intersection form on the Seifert surface, then
20:
1814:
1799:
1754:
1651:
1604:
1599:
1594:
1424:
1321:
853:
724:{\displaystyle \Delta (t)=c_{0}+c_{1}t+\cdots +c_{n}t^{n}+\cdots +c_{0}t^{2n}}
1819:
1487:
1159:. Annals of Mathematics Studies. Vol. 115. Princeton University Press.
28:
1804:
1414:
275:{\displaystyle \sum \limits _{i=1}^{g}v_{2i-1,2i-1}v_{2i,2i}{\pmod {2}}.}
1824:
1472:
1432:
576:
840:
Kunio
Murasugi proved that the Arf invariant is zero if and only if
1713:
609:
This approach to the Arf invariant is by
Raymond Robertello. Let
1784:
1137:. Mathematical notes. Vol. 30. Princeton University Press.
1203:
70:
of this quadratic form is the Arf invariant of the knot.
987:
with integer coefficients, we know that the determinant
572:
if they are related by a finite sequence of pass-moves.
738:
of the knot. Then the Arf invariant is the residue of
1104:
Robertello, Raymond, An
Invariant of Knot Corbordism,
1035:
993:
964:
896:
861:
747:
618:
528:
358:
291:
175:
84:
1742:
1686:
1521:
1423:
1388:
1246:
951:{\displaystyle \Delta (t)=p(t)p\left(t^{-1}\right)}
121:of the knot, constructed from a set of curves on a
1063:
1021:
979:
950:
882:
804:
723:
541:
507:
341:
274:
109:
1181:. Lecture Notes in Mathematics. Vol. 1374.
35:obtained from a quadratic form associated to a
1106:Communications on Pure and Applied Mathematics
1215:
805:{\displaystyle c_{n-1}+c_{n-3}+\cdots +c_{r}}
561:This approach to the Arf invariant is due to
8:
575:Every knot is pass-equivalent to either the
318:
292:
342:{\displaystyle \{a_{i},b_{i}\},i=1\ldots g}
1222:
1208:
1200:
1034:
992:
963:
935:
895:
883:{\displaystyle K\subset \mathbb {S} ^{3}}
874:
870:
869:
860:
796:
771:
752:
746:
712:
702:
683:
673:
651:
638:
617:
533:
527:
486:
475:
470:
457:
431:
426:
413:
392:
381:
357:
312:
299:
290:
253:
235:
201:
191:
180:
174:
95:
83:
43:is a Seifert surface of a knot, then the
1064:{\displaystyle \left|\Delta (-1)\right|}
1029:of a slice knot is a square integer. As
1022:{\displaystyle \left|\Delta (-1)\right|}
1079:
129:which represent a basis for the first
7:
1855:
1108:, Volume 18, pp. 543–555, 1965
16:Knot invariant named after Cahit Arf
494:
378:
261:
177:
1041:
999:
897:
619:
605:Definition by Alexander polynomial
14:
848:Arf as knot concordance invariant
133:of the surface. This means that
1854:
1843:
1842:
590:Definition by partition function
549:denotes the positive pushoff of
487:
254:
166:of the knot is the residue of
1709:Dowker–Thistlethwaite notation
1053:
1044:
1011:
1002:
974:
968:
921:
915:
906:
900:
628:
622:
557:Definition by pass equivalence
498:
488:
371:
365:
265:
255:
148:matrix with the property that
19:In the mathematical field of
1:
74:Definition by Seifert matrix
1179:The topology of 4-manifolds
1899:
568:We define two knots to be
1838:
1699:Alexander–Briggs notation
110:{\displaystyle V=v_{i,j}}
1095:Kauffman (1987) pp.75–78
1790:List of knots and links
1338:Kinoshita–Terasaka knot
27:of a knot, named after
1065:
1023:
981:
952:
884:
806:
725:
543:
509:
397:
343:
276:
196:
111:
1580:Finite type invariant
1066:
1024:
982:
953:
885:
807:
726:
544:
542:{\displaystyle a^{+}}
510:
377:
344:
277:
176:
112:
1086:Kauffman (1987) p.74
1033:
991:
980:{\displaystyle p(t)}
962:
958:for some polynomial
894:
859:
745:
736:Alexander polynomial
616:
526:
356:
289:
173:
82:
1750:Alexander's theorem
842:Δ(−1) ≡ ±1 modulo 8
480:
436:
1153:Kauffman, Louis H.
1135:Formal knot theory
1131:Kauffman, Louis H.
1061:
1019:
977:
948:
880:
802:
721:
539:
505:
466:
422:
339:
272:
107:
1870:
1869:
1724:Reidemeister move
1590:Khovanov homology
1585:Hyperbolic volume
285:Specifically, if
160:symplectic matrix
1890:
1858:
1857:
1846:
1845:
1810:Tait conjectures
1513:
1512:
1498:
1497:
1483:
1482:
1375:
1374:
1360:
1359:
1344:(−2,3,7) pretzel
1224:
1217:
1210:
1201:
1196:
1170:
1148:
1118:
1115:
1109:
1102:
1096:
1093:
1087:
1084:
1070:
1068:
1067:
1062:
1060:
1056:
1028:
1026:
1025:
1020:
1018:
1014:
986:
984:
983:
978:
957:
955:
954:
949:
947:
943:
942:
889:
887:
886:
881:
879:
878:
873:
843:
832:
821:
815:modulo 2, where
811:
809:
808:
803:
801:
800:
782:
781:
763:
762:
730:
728:
727:
722:
720:
719:
707:
706:
688:
687:
678:
677:
656:
655:
643:
642:
548:
546:
545:
540:
538:
537:
518:where lk is the
514:
512:
511:
506:
501:
485:
481:
479:
474:
462:
461:
441:
437:
435:
430:
418:
417:
396:
391:
348:
346:
345:
340:
317:
316:
304:
303:
281:
279:
278:
273:
268:
252:
251:
230:
229:
195:
190:
157:
147:
116:
114:
113:
108:
106:
105:
65:
1898:
1897:
1893:
1892:
1891:
1889:
1888:
1887:
1883:Knot invariants
1873:
1872:
1871:
1866:
1834:
1738:
1704:Conway notation
1688:
1682:
1669:Tricolorability
1517:
1511:
1508:
1507:
1506:
1496:
1493:
1492:
1491:
1481:
1478:
1477:
1476:
1468:
1458:
1448:
1438:
1419:
1398:Composite knots
1384:
1373:
1370:
1369:
1368:
1365:Borromean rings
1358:
1355:
1354:
1353:
1327:
1317:
1307:
1297:
1289:
1281:
1271:
1261:
1242:
1228:
1193:
1183:Springer-Verlag
1173:
1167:
1151:
1145:
1129:
1126:
1121:
1116:
1112:
1103:
1099:
1094:
1090:
1085:
1081:
1077:
1040:
1036:
1031:
1030:
998:
994:
989:
988:
960:
959:
931:
927:
892:
891:
868:
857:
856:
850:
841:
827:
816:
792:
767:
748:
743:
742:
708:
698:
679:
669:
647:
634:
614:
613:
607:
592:
570:pass equivalent
559:
529:
524:
523:
453:
452:
448:
409:
408:
404:
354:
353:
308:
295:
287:
286:
231:
197:
171:
170:
149:
138:
123:Seifert surface
91:
80:
79:
76:
51:
47:
37:Seifert surface
17:
12:
11:
5:
1896:
1894:
1886:
1885:
1875:
1874:
1868:
1867:
1865:
1864:
1852:
1839:
1836:
1835:
1833:
1832:
1830:Surgery theory
1827:
1822:
1817:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1757:
1752:
1746:
1744:
1740:
1739:
1737:
1736:
1731:
1729:Skein relation
1726:
1721:
1716:
1711:
1706:
1701:
1695:
1693:
1684:
1683:
1681:
1680:
1674:Unknotting no.
1671:
1666:
1661:
1660:
1659:
1649:
1644:
1643:
1642:
1637:
1632:
1627:
1622:
1612:
1607:
1602:
1597:
1592:
1587:
1582:
1577:
1572:
1567:
1566:
1565:
1555:
1550:
1549:
1548:
1538:
1533:
1527:
1525:
1519:
1518:
1516:
1515:
1509:
1500:
1494:
1485:
1479:
1470:
1466:
1460:
1456:
1450:
1446:
1440:
1436:
1429:
1427:
1421:
1420:
1418:
1417:
1412:
1411:
1410:
1405:
1394:
1392:
1386:
1385:
1383:
1382:
1377:
1371:
1362:
1356:
1347:
1341:
1335:
1329:
1325:
1319:
1315:
1309:
1305:
1299:
1295:
1291:
1287:
1283:
1279:
1273:
1269:
1263:
1259:
1252:
1250:
1244:
1243:
1229:
1227:
1226:
1219:
1212:
1204:
1198:
1197:
1191:
1171:
1165:
1149:
1143:
1125:
1122:
1120:
1119:
1110:
1097:
1088:
1078:
1076:
1073:
1059:
1055:
1052:
1049:
1046:
1043:
1039:
1017:
1013:
1010:
1007:
1004:
1001:
997:
976:
973:
970:
967:
946:
941:
938:
934:
930:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
877:
872:
867:
864:
849:
846:
813:
812:
799:
795:
791:
788:
785:
780:
777:
774:
770:
766:
761:
758:
755:
751:
732:
731:
718:
715:
711:
705:
701:
697:
694:
691:
686:
682:
676:
672:
668:
665:
662:
659:
654:
650:
646:
641:
637:
633:
630:
627:
624:
621:
606:
603:
591:
588:
563:Louis Kauffman
558:
555:
536:
532:
516:
515:
504:
500:
497:
493:
490:
484:
478:
473:
469:
465:
460:
456:
451:
447:
444:
440:
434:
429:
425:
421:
416:
412:
407:
403:
400:
395:
390:
387:
384:
380:
376:
373:
370:
367:
364:
361:
338:
335:
332:
329:
326:
323:
320:
315:
311:
307:
302:
298:
294:
283:
282:
271:
267:
264:
260:
257:
250:
247:
244:
241:
238:
234:
228:
225:
222:
219:
216:
213:
210:
207:
204:
200:
194:
189:
186:
183:
179:
119:Seifert matrix
104:
101:
98:
94:
90:
87:
75:
72:
49:
45:homology group
33:knot invariant
15:
13:
10:
9:
6:
4:
3:
2:
1895:
1884:
1881:
1880:
1878:
1863:
1862:
1853:
1851:
1850:
1841:
1840:
1837:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1765:Conway sphere
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1747:
1745:
1741:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1696:
1694:
1692:
1685:
1679:
1675:
1672:
1670:
1667:
1665:
1662:
1658:
1655:
1654:
1653:
1650:
1648:
1645:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1617:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1564:
1561:
1560:
1559:
1556:
1554:
1551:
1547:
1544:
1543:
1542:
1539:
1537:
1536:Arf invariant
1534:
1532:
1529:
1528:
1526:
1524:
1520:
1504:
1501:
1489:
1486:
1474:
1471:
1464:
1461:
1454:
1451:
1444:
1441:
1434:
1431:
1430:
1428:
1426:
1422:
1416:
1413:
1409:
1406:
1404:
1401:
1400:
1399:
1396:
1395:
1393:
1391:
1387:
1381:
1378:
1366:
1363:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1323:
1320:
1313:
1310:
1303:
1300:
1298:
1292:
1290:
1284:
1277:
1274:
1267:
1264:
1257:
1254:
1253:
1251:
1249:
1245:
1240:
1236:
1232:
1225:
1220:
1218:
1213:
1211:
1206:
1205:
1202:
1194:
1192:0-387-51148-2
1188:
1184:
1180:
1176:
1175:Kirby, Robion
1172:
1168:
1166:0-691-08435-1
1162:
1158:
1154:
1150:
1146:
1144:0-691-08336-3
1140:
1136:
1132:
1128:
1127:
1123:
1114:
1111:
1107:
1101:
1098:
1092:
1089:
1083:
1080:
1074:
1072:
1057:
1050:
1047:
1037:
1015:
1008:
1005:
995:
971:
965:
944:
939:
936:
932:
928:
924:
918:
912:
909:
903:
875:
865:
862:
855:
847:
845:
838:
836:
830:
825:
819:
797:
793:
789:
786:
783:
778:
775:
772:
768:
764:
759:
756:
753:
749:
741:
740:
739:
737:
716:
713:
709:
703:
699:
695:
692:
689:
684:
680:
674:
670:
666:
663:
660:
657:
652:
648:
644:
639:
635:
631:
625:
612:
611:
610:
604:
602:
600:
596:
595:Vaughan Jones
589:
587:
584:
582:
578:
573:
571:
566:
564:
556:
554:
552:
534:
530:
521:
502:
495:
491:
482:
476:
471:
467:
463:
458:
454:
449:
445:
442:
438:
432:
427:
423:
419:
414:
410:
405:
401:
398:
393:
388:
385:
382:
374:
368:
362:
359:
352:
351:
350:
336:
333:
330:
327:
324:
321:
313:
309:
305:
300:
296:
269:
262:
258:
248:
245:
242:
239:
236:
232:
226:
223:
220:
217:
214:
211:
208:
205:
202:
198:
192:
187:
184:
181:
169:
168:
167:
165:
164:Arf invariant
161:
156:
152:
146:
142:
136:
132:
128:
124:
120:
102:
99:
96:
92:
88:
85:
73:
71:
69:
68:Arf invariant
63:
59:
55:
46:
42:
38:
34:
30:
26:
25:Arf invariant
22:
1859:
1847:
1775:Double torus
1760:Braid theory
1575:Crossing no.
1570:Crosscap no.
1535:
1256:Figure-eight
1178:
1156:
1134:
1113:
1100:
1091:
1082:
851:
839:
834:
828:
823:
817:
814:
733:
608:
599:knot diagram
593:
585:
574:
569:
567:
560:
550:
517:
284:
163:
154:
150:
144:
140:
134:
126:
77:
61:
57:
53:
40:
24:
18:
1610:Linking no.
1531:Alternating
1332:Conway knot
1312:Carrick mat
1266:Three-twist
1231:Knot theory
890:factors as
520:link number
21:knot theory
1770:Complement
1734:Tabulation
1691:operations
1615:Polynomial
1605:Link group
1600:Knot group
1563:Invertible
1541:Bridge no.
1523:Invariants
1453:Cinquefoil
1322:Perko pair
1248:Hyperbolic
1124:References
854:slice knot
1664:Stick no.
1620:Alexander
1558:Chirality
1503:Solomon's
1463:Septafoil
1390:Satellite
1350:Whitehead
1276:Stevedore
1048:−
1042:Δ
1006:−
1000:Δ
937:−
898:Δ
866:⊂
826:odd, and
787:⋯
776:−
757:−
693:⋯
664:⋯
620:Δ
446:
402:
379:∑
363:
334:…
224:−
209:−
178:∑
125:of genus
29:Cahit Arf
1877:Category
1849:Category
1719:Mutation
1687:Notation
1640:Kauffman
1553:Brunnian
1546:2-bridge
1415:Knot sum
1346:(12n242)
1177:(1989).
1157:On knots
1155:(1987).
1133:(1983).
131:homology
1861:Commons
1780:Fibered
1678:problem
1647:Pretzel
1625:Bracket
1443:Trefoil
1380:L10a140
1340:(11n42)
1334:(11n34)
1302:Endless
734:be the
581:trefoil
579:or the
31:, is a
1825:Writhe
1795:Ribbon
1630:HOMFLY
1473:Unlink
1433:Unknot
1408:Square
1403:Granny
1189:
1163:
1141:
837:even.
577:unknot
162:. The
39:. If
23:, the
1815:Twist
1800:Slice
1755:Berge
1743:Other
1714:Flype
1652:Prime
1635:Jones
1595:Genus
1425:Torus
1239:links
1235:knots
1075:Notes
158:is a
137:is a
117:be a
1820:Wild
1785:Knot
1689:and
1676:and
1657:list
1488:Hopf
1237:and
1187:ISBN
1161:ISBN
1139:ISBN
833:for
822:for
565:.
522:and
78:Let
1805:Sum
1326:161
1324:(10
831:= 1
820:= 0
492:mod
360:Arf
259:mod
143:× 2
1879::
1505:(4
1490:(2
1475:(0
1465:(7
1455:(5
1445:(3
1435:(0
1367:(6
1352:(5
1316:18
1314:(8
1304:(7
1278:(6
1268:(5
1258:(4
1185:.
844:.
601:.
553:.
443:lk
399:lk
153:−
60:/2
56:,
1514:)
1510:1
1499:)
1495:1
1484:)
1480:1
1469:)
1467:1
1459:)
1457:1
1449:)
1447:1
1439:)
1437:1
1376:)
1372:2
1361:)
1357:1
1328:)
1318:)
1308:)
1306:4
1296:3
1294:6
1288:2
1286:6
1282:)
1280:1
1272:)
1270:2
1262:)
1260:1
1241:)
1233:(
1223:e
1216:t
1209:v
1195:.
1169:.
1147:.
1058:|
1054:)
1051:1
1045:(
1038:|
1016:|
1012:)
1009:1
1003:(
996:|
975:)
972:t
969:(
966:p
945:)
940:1
933:t
929:(
925:p
922:)
919:t
916:(
913:p
910:=
907:)
904:t
901:(
876:3
871:S
863:K
835:n
829:r
824:n
818:r
798:r
794:c
790:+
784:+
779:3
773:n
769:c
765:+
760:1
754:n
750:c
717:n
714:2
710:t
704:0
700:c
696:+
690:+
685:n
681:t
675:n
671:c
667:+
661:+
658:t
653:1
649:c
645:+
640:0
636:c
632:=
629:)
626:t
623:(
551:a
535:+
531:a
503:.
499:)
496:2
489:(
483:)
477:+
472:i
468:b
464:,
459:i
455:b
450:(
439:)
433:+
428:i
424:a
420:,
415:i
411:a
406:(
394:g
389:1
386:=
383:i
375:=
372:)
369:K
366:(
337:g
331:1
328:=
325:i
322:,
319:}
314:i
310:b
306:,
301:i
297:a
293:{
270:.
266:)
263:2
256:(
249:i
246:2
243:,
240:i
237:2
233:v
227:1
221:i
218:2
215:,
212:1
206:i
203:2
199:v
193:g
188:1
185:=
182:i
155:V
151:V
145:g
141:g
139:2
135:V
127:g
103:j
100:,
97:i
93:v
89:=
86:V
64:)
62:Z
58:Z
54:F
52:(
50:1
48:H
41:F
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