74:. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part of the disk through the slit. More precisely, this type of singularity is a closed arc consisting of intersection points of the disk with itself, such that the preimage of this arc consists of two arcs in the disc, one completely in the interior of the disk and the other having its two endpoints on the disk boundary.
1424:
20:
1436:
494:
suggested that the conjecture might not be true, and provided a family of knots that could be counterexamples to it. The conjecture was further strengthened when a famous potential counter-example, the (2, 1) cable of the figure-eight knot, was shown to be not slice and thereby not a counterexample.
196:
404:
336:
450:
237:
721:
Dai, Irving; Kang, Sungkyung; Mallick, Abhishek; Park, JungHwan; Stoffregen, Matthew (2022-07-28). "The $ (2,1)$ -cable of the figure-eight knot is not smoothly slice".
48:
138:
111:
801:
768:
1369:
1288:
743:
642:
143:
835:
1472:
1278:
357:
1283:
1154:
855:
242:
917:
409:
201:
923:
987:
982:
794:
1467:
1115:
1440:
1329:
1298:
1159:
536:(2010), "Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures",
583:
Greene, Joshua; Jabuka, Stanislav (2011), "The slice-ribbon conjecture for 3-stranded pretzel knots",
1462:
1428:
1199:
787:
1236:
1219:
1257:
1204:
818:
814:
722:
695:
669:
618:
592:
571:
545:
529:
67:
1354:
1303:
1253:
1209:
1169:
1164:
1082:
638:
1389:
1214:
1110:
845:
679:
602:
555:
533:
26:
691:
652:
637:, Annals of Mathematics Studies, vol. 115, Princeton, NJ: Princeton University Press,
614:
567:
517:
116:
89:
1349:
1313:
1248:
1194:
1149:
1142:
1032:
944:
827:
748:
687:
648:
610:
563:
513:
1409:
1308:
1270:
1189:
1102:
977:
969:
929:
630:
510:
Topology of 3-manifolds and related topics (Proc. The Univ. of
Georgia Institute, 1961)
347:
1456:
1344:
1132:
1125:
1120:
479:
699:
622:
575:
1359:
1339:
1243:
1226:
1022:
959:
525:
487:
1042:
881:
873:
865:
1137:
911:
891:
810:
779:
59:
55:
19:
1394:
1379:
1334:
1231:
1184:
1179:
1174:
1004:
901:
660:
Lisca, Paolo (2007), "Lens spaces, rational balls and the ribbon conjecture",
461:
1399:
1067:
559:
505:
465:
683:
606:
1384:
994:
472:, asks if the converse is true: is every (smoothly) slice knot ribbon?
1404:
1052:
1012:
674:
769:"How Complex Is a Knot? New Proof Reveals Ranking System That Works"
727:
1293:
744:"Mathematicians Eliminate Long-Standing Threat to Knot Conjecture"
597:
550:
18:
1364:
783:
191:{\displaystyle M\cap \partial D^{4}=\partial M\subset S^{3}}
512:, Englewood Cliffs, N.J.: Prentice-Hall, pp. 168–176,
491:
399:{\displaystyle \partial M\subset \partial D^{4}=S^{3}}
412:
360:
245:
204:
146:
119:
92:
29:
1322:
1266:
1101:
1003:
968:
826:
444:
398:
331:{\displaystyle f(x,y,z,w)=x^{2}+y^{2}+z^{2}+w^{2}}
330:
231:
190:
132:
105:
42:
478:showed that the conjecture is true for knots of
70:that bounds a self-intersecting disk with only
445:{\displaystyle f_{|M}\colon M\to \mathbb {R} }
795:
232:{\displaystyle f\colon D^{4}\to \mathbb {R} }
23:A 3-dimensional rendering of the ribbon knot
8:
483:
802:
788:
780:
726:
673:
596:
549:
438:
437:
418:
417:
411:
390:
377:
359:
322:
309:
296:
283:
244:
225:
224:
215:
203:
182:
160:
145:
124:
118:
97:
91:
34:
28:
508:(1962), "Some problems in knot theory",
492:Gompf, Scharlemann & Thompson (2010)
486:showed it to be true for three-stranded
713:
475:
7:
1435:
464:. A famous open problem, posed by
460:Every ribbon knot is known to be a
370:
361:
169:
153:
14:
742:Sloman, Leila (2 February 2023).
521:. Reprinted by Dover Books, 2010.
1434:
1423:
1422:
585:American Journal of Mathematics
1289:Dowker–Thistlethwaite notation
490:with odd parameters. However,
452:has no interior local maxima.
434:
419:
273:
249:
221:
1:
50:, showing the ribbon property
767:Sloman, Leila (2022-05-18).
78:Morse-theoretic formulation
1489:
484:Greene & Jabuka (2011)
1418:
1279:Alexander–Briggs notation
338:. By a small isotopy of
198:. Consider the function
16:Type of mathematical knot
1370:List of knots and links
918:Kinoshita–Terasaka knot
662:Geometry & Topology
560:10.2140/gt.2010.14.2305
538:Geometry & Topology
470:slice-ribbon conjecture
456:Slice-ribbon conjecture
86:is a smoothly embedded
684:10.2140/gt.2007.11.429
446:
400:
332:
233:
192:
134:
107:
51:
44:
43:{\displaystyle 8_{20}}
1473:Slice knots and links
1160:Finite type invariant
607:10.1353/ajm.2011.0022
447:
401:
333:
234:
193:
135:
133:{\displaystyle D^{4}}
108:
106:{\displaystyle D^{2}}
45:
22:
410:
406:is a ribbon knot if
358:
342:one can ensure that
243:
202:
144:
117:
90:
72:ribbon singularities
27:
1330:Alexander's theorem
530:Scharlemann, Martin
631:Kauffman, Louis H.
442:
396:
328:
229:
188:
130:
103:
52:
40:
1450:
1449:
1304:Reidemeister move
1170:Khovanov homology
1165:Hyperbolic volume
534:Thompson, Abigail
468:and known as the
1480:
1438:
1437:
1426:
1425:
1390:Tait conjectures
1093:
1092:
1078:
1077:
1063:
1062:
955:
954:
940:
939:
924:(−2,3,7) pretzel
804:
797:
790:
781:
776:
754:
753:
739:
733:
732:
730:
718:
702:
677:
655:
625:
600:
578:
553:
544:(4): 2305–2347,
526:Gompf, Robert E.
520:
451:
449:
448:
443:
441:
427:
426:
422:
405:
403:
402:
397:
395:
394:
382:
381:
337:
335:
334:
329:
327:
326:
314:
313:
301:
300:
288:
287:
238:
236:
235:
230:
228:
220:
219:
197:
195:
194:
189:
187:
186:
165:
164:
139:
137:
136:
131:
129:
128:
112:
110:
109:
104:
102:
101:
49:
47:
46:
41:
39:
38:
1488:
1487:
1483:
1482:
1481:
1479:
1478:
1477:
1468:Knots and links
1453:
1452:
1451:
1446:
1414:
1318:
1284:Conway notation
1268:
1262:
1249:Tricolorability
1097:
1091:
1088:
1087:
1086:
1076:
1073:
1072:
1071:
1061:
1058:
1057:
1056:
1048:
1038:
1028:
1018:
999:
978:Composite knots
964:
953:
950:
949:
948:
945:Borromean rings
938:
935:
934:
933:
907:
897:
887:
877:
869:
861:
851:
841:
822:
808:
773:Quanta Magazine
766:
763:
758:
757:
749:Quanta Magazine
741:
740:
736:
720:
719:
715:
710:
659:
645:
629:
582:
524:
504:
501:
458:
413:
408:
407:
386:
373:
356:
355:
346:restricts to a
318:
305:
292:
279:
241:
240:
211:
200:
199:
178:
156:
142:
141:
120:
115:
114:
93:
88:
87:
80:
30:
25:
24:
17:
12:
11:
5:
1486:
1484:
1476:
1475:
1470:
1465:
1455:
1454:
1448:
1447:
1445:
1444:
1432:
1419:
1416:
1415:
1413:
1412:
1410:Surgery theory
1407:
1402:
1397:
1392:
1387:
1382:
1377:
1372:
1367:
1362:
1357:
1352:
1347:
1342:
1337:
1332:
1326:
1324:
1320:
1319:
1317:
1316:
1311:
1309:Skein relation
1306:
1301:
1296:
1291:
1286:
1281:
1275:
1273:
1264:
1263:
1261:
1260:
1254:Unknotting no.
1251:
1246:
1241:
1240:
1239:
1229:
1224:
1223:
1222:
1217:
1212:
1207:
1202:
1192:
1187:
1182:
1177:
1172:
1167:
1162:
1157:
1152:
1147:
1146:
1145:
1135:
1130:
1129:
1128:
1118:
1113:
1107:
1105:
1099:
1098:
1096:
1095:
1089:
1080:
1074:
1065:
1059:
1050:
1046:
1040:
1036:
1030:
1026:
1020:
1016:
1009:
1007:
1001:
1000:
998:
997:
992:
991:
990:
985:
974:
972:
966:
965:
963:
962:
957:
951:
942:
936:
927:
921:
915:
909:
905:
899:
895:
889:
885:
879:
875:
871:
867:
863:
859:
853:
849:
843:
839:
832:
830:
824:
823:
809:
807:
806:
799:
792:
784:
778:
777:
762:
761:External links
759:
756:
755:
734:
712:
711:
709:
706:
705:
704:
657:
643:
627:
591:(3): 555–580,
580:
522:
500:
497:
457:
454:
440:
436:
433:
430:
425:
421:
416:
393:
389:
385:
380:
376:
372:
369:
366:
363:
348:Morse function
325:
321:
317:
312:
308:
304:
299:
295:
291:
286:
282:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
227:
223:
218:
214:
210:
207:
185:
181:
177:
174:
171:
168:
163:
159:
155:
152:
149:
127:
123:
100:
96:
79:
76:
37:
33:
15:
13:
10:
9:
6:
4:
3:
2:
1485:
1474:
1471:
1469:
1466:
1464:
1461:
1460:
1458:
1443:
1442:
1433:
1431:
1430:
1421:
1420:
1417:
1411:
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1346:
1345:Conway sphere
1343:
1341:
1338:
1336:
1333:
1331:
1328:
1327:
1325:
1321:
1315:
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1280:
1277:
1276:
1274:
1272:
1265:
1259:
1255:
1252:
1250:
1247:
1245:
1242:
1238:
1235:
1234:
1233:
1230:
1228:
1225:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1203:
1201:
1198:
1197:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1144:
1141:
1140:
1139:
1136:
1134:
1131:
1127:
1124:
1123:
1122:
1119:
1117:
1116:Arf invariant
1114:
1112:
1109:
1108:
1106:
1104:
1100:
1084:
1081:
1069:
1066:
1054:
1051:
1044:
1041:
1034:
1031:
1024:
1021:
1014:
1011:
1010:
1008:
1006:
1002:
996:
993:
989:
986:
984:
981:
980:
979:
976:
975:
973:
971:
967:
961:
958:
946:
943:
931:
928:
925:
922:
919:
916:
913:
910:
903:
900:
893:
890:
883:
880:
878:
872:
870:
864:
857:
854:
847:
844:
837:
834:
833:
831:
829:
825:
820:
816:
812:
805:
800:
798:
793:
791:
786:
785:
782:
774:
770:
765:
764:
760:
751:
750:
745:
738:
735:
729:
724:
717:
714:
707:
701:
697:
693:
689:
685:
681:
676:
671:
667:
663:
658:
654:
650:
646:
644:0-691-08434-3
640:
636:
632:
628:
624:
620:
616:
612:
608:
604:
599:
594:
590:
586:
581:
577:
573:
569:
565:
561:
557:
552:
547:
543:
539:
535:
531:
527:
523:
519:
515:
511:
507:
503:
502:
498:
496:
493:
489:
488:pretzel knots
485:
481:
480:bridge number
477:
473:
471:
467:
463:
455:
453:
431:
428:
423:
414:
391:
387:
383:
378:
374:
367:
364:
353:
349:
345:
341:
323:
319:
315:
310:
306:
302:
297:
293:
289:
284:
280:
276:
270:
267:
264:
261:
258:
255:
252:
246:
216:
212:
208:
205:
183:
179:
175:
172:
166:
161:
157:
150:
147:
125:
121:
98:
94:
85:
82:A slice disc
77:
75:
73:
69:
65:
61:
57:
35:
31:
21:
1439:
1427:
1374:
1355:Double torus
1340:Braid theory
1155:Crossing no.
1150:Crosscap no.
836:Figure-eight
772:
747:
737:
716:
675:math/0701610
665:
661:
634:
588:
584:
541:
537:
509:
476:Lisca (2007)
474:
469:
459:
354:. One says
351:
343:
339:
83:
81:
71:
63:
56:mathematical
53:
1463:Knot theory
1190:Linking no.
1111:Alternating
912:Conway knot
892:Carrick mat
846:Three-twist
811:Knot theory
668:: 429–472,
64:ribbon knot
60:knot theory
1457:Categories
1350:Complement
1314:Tabulation
1271:operations
1195:Polynomial
1185:Link group
1180:Knot group
1143:Invertible
1121:Bridge no.
1103:Invariants
1033:Cinquefoil
902:Perko pair
828:Hyperbolic
728:2207.14187
708:References
506:Fox, R. H.
499:References
462:slice knot
1244:Stick no.
1200:Alexander
1138:Chirality
1083:Solomon's
1043:Septafoil
970:Satellite
930:Whitehead
856:Stevedore
598:0706.3398
551:1103.1601
466:Ralph Fox
435:→
429::
371:∂
368:⊂
362:∂
239:given by
222:→
209::
176:⊂
170:∂
154:∂
151:∩
1429:Category
1299:Mutation
1267:Notation
1220:Kauffman
1133:Brunnian
1126:2-bridge
995:Knot sum
926:(12n242)
700:15238217
635:On Knots
633:(1987),
623:10279100
576:58915479
58:area of
1441:Commons
1360:Fibered
1258:problem
1227:Pretzel
1205:Bracket
1023:Trefoil
960:L10a140
920:(11n42)
914:(11n34)
882:Endless
692:2302495
653:0907872
615:2808326
568:2740649
518:0140100
54:In the
1405:Writhe
1375:Ribbon
1210:HOMFLY
1053:Unlink
1013:Unknot
988:Square
983:Granny
698:
690:
651:
641:
621:
613:
574:
566:
516:
1395:Twist
1380:Slice
1335:Berge
1323:Other
1294:Flype
1232:Prime
1215:Jones
1175:Genus
1005:Torus
819:links
815:knots
723:arXiv
696:S2CID
670:arXiv
619:S2CID
593:arXiv
572:S2CID
546:arXiv
482:two.
140:with
66:is a
1400:Wild
1365:Knot
1269:and
1256:and
1237:list
1068:Hopf
817:and
639:ISBN
68:knot
62:, a
1385:Sum
906:161
904:(10
680:doi
603:doi
589:133
556:doi
350:on
113:in
1459::
1085:(4
1070:(2
1055:(0
1045:(7
1035:(5
1025:(3
1015:(0
947:(6
932:(5
896:18
894:(8
884:(7
858:(6
848:(5
838:(4
771:.
746:.
694:,
688:MR
686:,
678:,
666:11
664:,
649:MR
647:,
617:,
611:MR
609:,
601:,
587:,
570:,
564:MR
562:,
554:,
542:14
540:,
532:;
528:;
514:MR
36:20
1094:)
1090:1
1079:)
1075:1
1064:)
1060:1
1049:)
1047:1
1039:)
1037:1
1029:)
1027:1
1019:)
1017:1
956:)
952:2
941:)
937:1
908:)
898:)
888:)
886:4
876:3
874:6
868:2
866:6
862:)
860:1
852:)
850:2
842:)
840:1
821:)
813:(
803:e
796:t
789:v
775:.
752:.
731:.
725::
703:.
682::
672::
656:.
626:.
605::
595::
579:.
558::
548::
439:R
432:M
424:M
420:|
415:f
392:3
388:S
384:=
379:4
375:D
365:M
352:M
344:f
340:M
324:2
320:w
316:+
311:2
307:z
303:+
298:2
294:y
290:+
285:2
281:x
277:=
274:)
271:w
268:,
265:z
262:,
259:y
256:,
253:x
250:(
247:f
226:R
217:4
213:D
206:f
184:3
180:S
173:M
167:=
162:4
158:D
148:M
126:4
122:D
99:2
95:D
84:M
32:8
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
