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It can be difficult to find a way to untangle string even though the fact it started out untangled proves the task is possible. Thistlethwaite and Ochiai provided many examples of diagrams of unknots that have no obvious way to simplify them, requiring one to temporarily increase the diagram's
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While rope is generally not in the form of a closed loop, sometimes there is a canonical way to imagine the ends being joined together. From this point of view, many useful practical knots are actually the unknot, including those that can be tied in a
503:
is a particular unknotted linkage that cannot be reconfigured into a flat convex polygon. Like crossing number, a linkage might need to be made more complex by subdividing its segments before it can be simplified.
600:
617:(both of which have 11 crossings) have the same Alexander and Conway polynomials as the unknot. It is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot.
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detect the unknot, but these are not known to be efficiently computable for this purpose. It is not known whether the Jones polynomial or
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105:
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855:
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495:, which is a collection of rigid line segments connected by universal joints at their endpoints. The
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248:
54:
1115:
423:
339:, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a
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39:
32:
1329:
1298:
307:
664: – Minimum number of times a specific knot must be passed through itself to become untied
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492:
8:
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79:
1502:
1389:
1214:
1110:
845:
517:
403:, since it was thought this approach would possibly give an efficient algorithm to
378:
1349:
1313:
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is the minimal number of segments needed to represent a knot as a linkage, and a
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Deciding if a particular knot is the unknot was a major driving force behind
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344:
652: – Embedding of the circle in three dimensional Euclidean space
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1384:
994:
382:
351:
276:
369:
The unknot is the only knot that is the boundary of an embedded
1404:
1052:
655:
359:
343:
tied into it, unknotted. To a knot theorist, an unknot is any
141:
130:
1293:
595:{\displaystyle \Delta (t)=1,\quad \nabla (z)=1,\quad V(q)=1.}
416:
658: – Link that consists of finitely many unlinked unknots
1364:
373:, which gives the characterization that only unknots have
315:
748:
529:
46:. Unsourced material may be challenged and removed.
707:
594:
809:
1454:
760:
358:(that is, deformable) to a geometrically round
795:
411:. Unknot recognition is known to be in both
802:
788:
609:has trivial Alexander polynomial, but the
129:
681:
106:Learn how and when to remove this message
314:
713:"A new class of stuck unknots in Pol-6"
1455:
783:
761:
720:Contributions to Algebra and Geometry
675:
388:
1435:
44:adding citations to reliable sources
15:
13:
620:The unknot is the only knot whose
552:
530:
14:
1514:
1493:Fully amphichiral knots and links
741:
407:from some presentation such as a
377:0. Similarly, the unknot is the
319:Two simple diagrams of the unknot
1498:Non-tricolorable knots and links
1434:
1423:
1422:
467:
452:
20:
1468:Non-alternating knots and links
605:No other knot with 10 or fewer
573:
551:
31:needs additional citations for
1289:Dowker–Thistlethwaite notation
701:
583:
577:
561:
555:
539:
533:
1:
757:. Accessed: May 7, 2013.
668:
507:
726:(2): 301–306. Archived from
325:mathematical theory of knots
7:
643:
520:of the unknot are trivial:
514:Alexander–Conway polynomial
437:
118:Loop seen as a trivial knot
10:
1519:
392:
1418:
1322:
1279:Alexander–Briggs notation
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1101:
1003:
968:
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128:
123:
491:can be represented as a
1478:Fibered knots and links
1370:List of knots and links
918:Kinoshita–Terasaka knot
611:Kinoshita–Terasaka knot
474:One of Ochiai's unknots
434:can detect the unknot.
596:
432:finite type invariants
320:
1488:Slice knots and links
1483:Prime knots and links
1473:Torus knots and links
1160:Finite type invariant
597:
318:
527:
405:recognize the unknot
381:with respect to the
40:improve this article
1330:Alexander's theorem
424:knot Floer homology
763:Weisstein, Eric W.
709:Godfried Toussaint
650:Knot (mathematics)
592:
395:Unknotting problem
389:Unknotting problem
348:topological circle
321:
1450:
1449:
1304:Reidemeister move
1170:Khovanov homology
1165:Hyperbolic volume
662:Unknotting number
428:Khovanov homology
422:It is known that
313:
312:
308:fully amphichiral
116:
115:
108:
90:
1510:
1438:
1437:
1426:
1425:
1390:Tait conjectures
1093:
1092:
1078:
1077:
1063:
1062:
955:
954:
940:
939:
924:(−2,3,7) pretzel
804:
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790:
781:
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775:
735:
734:
732:
717:
705:
699:
698:
696:
695:
686:. Archived from
679:
601:
599:
598:
593:
518:Jones polynomial
471:
456:
379:identity element
356:ambient isotopic
133:
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24:
16:
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1509:
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1507:
1453:
1452:
1451:
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1414:
1318:
1284:Conway notation
1268:
1262:
1249:Tricolorability
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978:Composite knots
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945:Borromean rings
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684:"Knotty topics"
682:Volker Schatz.
680:
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630:knot complement
624:is an infinite
528:
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510:
475:
472:
463:
457:
445:crossing number
440:
401:knot invariants
397:
391:
364:standard unknot
280:
262:Dowker notation
256:
239:Conway notation
119:
112:
101:
95:
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49:
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37:
25:
12:
11:
5:
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1413:
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1410:Surgery theory
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1402:
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1309:Skein relation
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1301:
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1261:
1260:
1254:Unknotting no.
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754:The Knot Atlas
743:
742:External links
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736:
733:on 2003-05-12.
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460:Thistlethwaite
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393:Main article:
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229:Unknotting no.
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1345:Conway sphere
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1123:
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1119:
1117:
1116:Arf invariant
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1112:
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1100:
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893:
890:
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844:
837:
834:
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831:
829:
825:
820:
816:
812:
805:
800:
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793:
791:
786:
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764:
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750:
746:
745:
729:
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721:
714:
710:
704:
690:on 2011-07-17
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631:
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567:
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542:
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502:
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494:
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485:
483:
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461:
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446:
435:
433:
429:
425:
420:
418:
414:
410:
406:
402:
396:
386:
384:
380:
376:
375:Seifert genus
372:
367:
365:
361:
357:
353:
349:
346:
342:
338:
334:
330:
326:
317:
309:
305:
301:
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293:
289:
284:
281:
275:
273:
269:
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240:
236:
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192:
190:
186:
182:
180:
176:
172:
170:
166:
162:
160:
156:
152:
150:
149:Arf invariant
146:
143:
140:
136:
132:
127:
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110:
107:
99:
96:November 2021
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
1439:
1427:
1355:Double torus
1340:Braid theory
1155:Crossing no.
1150:Crosscap no.
1012:
836:Figure-eight
769:
752:
728:the original
723:
719:
703:
692:. Retrieved
688:the original
677:
634:homeomorphic
626:cyclic group
619:
604:
511:
501:stuck unknot
497:stick number
486:
478:
441:
421:
409:knot diagram
398:
368:
363:
337:trivial knot
336:
332:
328:
322:
249:A–B notation
179:Crossing no.
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
1463:Knot theory
1190:Linking no.
1111:Alternating
912:Conway knot
892:Carrick mat
846:Three-twist
811:Knot theory
638:solid torus
615:Conway knot
385:operation.
199:Linking no.
138:Common name
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
694:2007-04-23
669:References
628:, and its
622:knot group
508:Invariants
219:Tunnel no.
169:Bridge no.
66:newspapers
1244:Stick no.
1200:Alexander
1138:Chirality
1083:Solomon's
1043:Septafoil
970:Satellite
930:Whitehead
856:Stevedore
771:MathWorld
607:crossings
553:∇
531:Δ
489:tame knot
209:Stick no.
159:Braid no.
1429:Category
1299:Mutation
1267:Notation
1220:Kauffman
1133:Brunnian
1126:2-bridge
995:Knot sum
926:(12n242)
766:"Unknot"
711:(2001).
644:See also
438:Examples
383:knot sum
354:that is
352:3-sphere
345:embedded
333:not knot
55:"Unknot"
1503:Circles
1441:Commons
1360:Fibered
1258:problem
1227:Pretzel
1205:Bracket
1023:Trefoil
960:L10a140
920:(11n42)
914:(11n34)
882:Endless
493:linkage
350:in the
323:In the
296:fibered
80:scholar
1405:Writhe
1375:Ribbon
1210:HOMFLY
1053:Unlink
1013:Unknot
988:Square
983:Granny
749:Unknot
656:Unlink
487:Every
462:unknot
362:, the
360:circle
329:unknot
327:, the
306:,
302:,
298:,
294:,
142:Circle
124:Unknot
82:
75:
68:
61:
53:
1395:Twist
1380:Slice
1335:Berge
1323:Other
1294:Flype
1232:Prime
1215:Jones
1175:Genus
1005:Torus
819:links
815:knots
731:(PDF)
716:(PDF)
636:to a
482:bight
417:co-NP
335:, or
304:slice
300:prime
292:torus
286:Other
189:Genus
87:JSTOR
73:books
1400:Wild
1365:Knot
1269:and
1256:and
1237:list
1068:Hopf
817:and
613:and
516:and
512:The
426:and
415:and
371:disk
341:knot
272:Next
59:news
1385:Sum
906:161
904:(10
751:",
632:is
42:by
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
768:.
724:42
722:.
718:.
640:.
590:1.
484:.
447:.
419:.
413:NP
366:.
331:,
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
774:.
747:"
697:.
587:=
584:)
581:q
578:(
575:V
571:,
568:1
565:=
562:)
559:z
556:(
549:,
546:1
543:=
540:)
537:t
534:(
279:1
277:3
266:-
255:1
253:0
243:-
233:0
223:0
213:3
203:0
193:0
183:0
173:0
163:1
153:0
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
