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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
490:
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
514:
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.
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628:(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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116:
97:
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69:
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1451:
866:
17:
928:
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524:
83:
506:, which is a collection of rigid line segments connected by universal joints at their endpoints. The
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998:
993:
934:
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350:, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a
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50:
43:
1340:
1309:
318:
675: – Minimum number of times a specific knot must be passed through itself to become untied
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1473:
1439:
1210:
798:
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503:
8:
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773:
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438:
239:
90:
1513:
1400:
1225:
1121:
856:
528:
414:, since it was thought this approach would possibly give an efficient algorithm to
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1360:
1324:
1259:
1205:
1160:
1153:
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640:
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is the minimal number of segments needed to represent a knot as a linkage, and a
423:
385:
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272:
790:
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1355:
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Deciding if a particular knot is the unknot was a major driving force behind
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1390:
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1185:
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1410:
1078:
781:
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355:
663: – Embedding of the circle in three dimensional Euclidean space
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1395:
1005:
393:
362:
287:
380:
The unknot is the only knot that is the boundary of an embedded
1415:
1063:
666:
370:
354:
tied into it, unknotted. To a knot theorist, an unknot is any
152:
141:
1304:
606:{\displaystyle \Delta (t)=1,\quad \nabla (z)=1,\quad V(q)=1.}
427:
669: – Link that consists of finitely many unlinked unknots
1375:
384:, which gives the characterization that only unknots have
326:
759:
540:
57:. Unsourced material may be challenged and removed.
718:
605:
820:
1465:
771:
369:(that is, deformable) to a geometrically round
806:
422:. Unknot recognition is known to be in both
813:
799:
620:has trivial Alexander polynomial, but the
140:
692:
117:Learn how and when to remove this message
325:
724:"A new class of stuck unknots in Pol-6"
14:
1466:
794:
772:
731:Contributions to Algebra and Geometry
686:
399:
1446:
55:adding citations to reliable sources
26:
24:
631:The unknot is the only knot whose
563:
541:
25:
1525:
1504:Fully amphichiral knots and links
752:
418:from some presentation such as a
388:0. Similarly, the unknot is the
330:Two simple diagrams of the unknot
1509:Non-tricolorable knots and links
1445:
1434:
1433:
478:
463:
31:
1479:Non-alternating knots and links
616:No other knot with 10 or fewer
584:
562:
42:needs additional citations for
1300:Dowker–Thistlethwaite notation
712:
594:
588:
572:
566:
550:
544:
13:
1:
768:. Accessed: May 7, 2013.
679:
518:
737:(2): 301–306. Archived from
336:mathematical theory of knots
7:
654:
531:of the unknot are trivial:
525:Alexander–Conway polynomial
448:
129:Loop seen as a trivial knot
10:
1530:
403:
1429:
1333:
1290:Alexander–Briggs notation
1277:
1112:
1014:
979:
837:
301:
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228:
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168:
158:
148:
139:
134:
502:can be represented as a
1489:Fibered knots and links
1381:List of knots and links
929:Kinoshita–Terasaka knot
622:Kinoshita–Terasaka knot
485:One of Ochiai's unknots
445:can detect the unknot.
607:
443:finite type invariants
331:
1499:Slice knots and links
1494:Prime knots and links
1484:Torus knots and links
1171:Finite type invariant
608:
329:
538:
416:recognize the unknot
392:with respect to the
51:improve this article
1341:Alexander's theorem
435:knot Floer homology
774:Weisstein, Eric W.
720:Godfried Toussaint
661:Knot (mathematics)
603:
406:Unknotting problem
400:Unknotting problem
359:topological circle
332:
1461:
1460:
1315:Reidemeister move
1181:Khovanov homology
1176:Hyperbolic volume
673:Unknotting number
439:Khovanov homology
433:It is known that
324:
323:
319:fully amphichiral
127:
126:
119:
101:
16:(Redirected from
1521:
1449:
1448:
1437:
1436:
1401:Tait conjectures
1104:
1103:
1089:
1088:
1074:
1073:
966:
965:
951:
950:
935:(−2,3,7) pretzel
815:
808:
801:
792:
791:
787:
786:
746:
745:
743:
728:
716:
710:
709:
707:
706:
697:. Archived from
690:
612:
610:
609:
604:
529:Jones polynomial
482:
467:
390:identity element
367:ambient isotopic
144:
132:
131:
122:
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111:
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100:
59:
35:
27:
21:
1529:
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1523:
1522:
1520:
1519:
1518:
1464:
1463:
1462:
1457:
1425:
1329:
1295:Conway notation
1279:
1273:
1260:Tricolorability
1108:
1102:
1099:
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1097:
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989:Composite knots
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956:Borromean rings
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713:
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695:"Knotty topics"
693:Volker Schatz.
691:
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682:
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641:knot complement
635:is an infinite
539:
536:
535:
521:
486:
483:
474:
468:
456:crossing number
451:
412:knot invariants
408:
402:
375:standard unknot
291:
273:Dowker notation
267:
250:Conway notation
130:
123:
112:
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60:
58:
48:
36:
23:
22:
15:
12:
11:
5:
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1506:
1501:
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1423:
1421:Surgery theory
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1320:Skein relation
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1312:
1307:
1302:
1297:
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1284:
1275:
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1265:Unknotting no.
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1234:
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1228:
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1218:
1213:
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1002:
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769:
765:The Knot Atlas
754:
753:External links
751:
748:
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744:on 2003-05-12.
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471:Thistlethwaite
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404:Main article:
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246:
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240:Unknotting no.
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1379:
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1359:
1357:
1356:Conway sphere
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1323:
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1151:
1150:
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1135:
1134:
1133:
1130:
1128:
1127:Arf invariant
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1123:
1120:
1119:
1117:
1115:
1111:
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1092:
1080:
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1065:
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868:
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836:
831:
827:
823:
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811:
809:
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802:
797:
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766:
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721:
715:
701:on 2011-07-17
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569:
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501:
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457:
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444:
440:
436:
431:
429:
425:
421:
417:
413:
407:
397:
395:
391:
387:
386:Seifert genus
383:
378:
376:
372:
368:
364:
360:
357:
353:
349:
345:
341:
337:
328:
320:
316:
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300:
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292:
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203:
201:
197:
193:
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187:
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181:
177:
173:
171:
167:
163:
161:
160:Arf invariant
157:
154:
151:
147:
143:
138:
133:
121:
118:
110:
107:November 2021
99:
96:
92:
89:
85:
82:
78:
75:
71:
68: –
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
1450:
1438:
1366:Double torus
1351:Braid theory
1166:Crossing no.
1161:Crosscap no.
1023:
847:Figure-eight
780:
763:
739:the original
734:
730:
714:
703:. Retrieved
699:the original
688:
645:homeomorphic
637:cyclic group
630:
615:
522:
512:stuck unknot
508:stick number
497:
489:
452:
432:
420:knot diagram
409:
379:
374:
348:trivial knot
347:
343:
339:
333:
260:A–B notation
190:Crossing no.
113:
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
18:Trivial knot
1474:Knot theory
1201:Linking no.
1122:Alternating
923:Conway knot
903:Carrick mat
857:Three-twist
822:Knot theory
649:solid torus
626:Conway knot
396:operation.
210:Linking no.
149:Common name
1468:Categories
1361:Complement
1325:Tabulation
1282:operations
1206:Polynomial
1196:Link group
1191:Knot group
1154:Invertible
1132:Bridge no.
1114:Invariants
1044:Cinquefoil
913:Perko pair
839:Hyperbolic
705:2007-04-23
680:References
639:, and its
633:knot group
519:Invariants
230:Tunnel no.
180:Bridge no.
77:newspapers
1255:Stick no.
1211:Alexander
1149:Chirality
1094:Solomon's
1054:Septafoil
981:Satellite
941:Whitehead
867:Stevedore
782:MathWorld
618:crossings
564:∇
542:Δ
500:tame knot
220:Stick no.
170:Braid no.
1440:Category
1310:Mutation
1278:Notation
1231:Kauffman
1144:Brunnian
1137:2-bridge
1006:Knot sum
937:(12n242)
777:"Unknot"
722:(2001).
655:See also
449:Examples
394:knot sum
365:that is
363:3-sphere
356:embedded
344:not knot
66:"Unknot"
1514:Circles
1452:Commons
1371:Fibered
1269:problem
1238:Pretzel
1216:Bracket
1034:Trefoil
971:L10a140
931:(11n42)
925:(11n34)
893:Endless
504:linkage
361:in the
334:In the
307:fibered
91:scholar
1416:Writhe
1386:Ribbon
1221:HOMFLY
1064:Unlink
1024:Unknot
999:Square
994:Granny
760:Unknot
667:Unlink
498:Every
473:unknot
373:, the
371:circle
340:unknot
338:, the
317:,
313:,
309:,
305:,
153:Circle
135:Unknot
93:
86:
79:
72:
64:
1406:Twist
1391:Slice
1346:Berge
1334:Other
1305:Flype
1243:Prime
1226:Jones
1186:Genus
1016:Torus
830:links
826:knots
742:(PDF)
727:(PDF)
647:to a
493:bight
428:co-NP
346:, or
315:slice
311:prime
303:torus
297:Other
200:Genus
98:JSTOR
84:books
1411:Wild
1376:Knot
1280:and
1267:and
1248:list
1079:Hopf
828:and
624:and
527:and
523:The
437:and
426:and
382:disk
352:knot
283:Next
70:news
1396:Sum
917:161
915:(10
762:",
643:is
53:by
1470::
1096:(4
1081:(2
1066:(0
1056:(7
1046:(5
1036:(3
1026:(0
958:(6
943:(5
907:18
905:(8
895:(7
869:(6
859:(5
849:(4
779:.
735:42
733:.
729:.
651:.
601:1.
495:.
458:.
430:.
424:NP
377:.
342:,
1105:)
1101:1
1090:)
1086:1
1075:)
1071:1
1060:)
1058:1
1050:)
1048:1
1040:)
1038:1
1030:)
1028:1
967:)
963:2
952:)
948:1
919:)
909:)
899:)
897:4
887:3
885:6
879:2
877:6
873:)
871:1
863:)
861:2
853:)
851:1
832:)
824:(
814:e
807:t
800:v
785:.
758:"
708:.
598:=
595:)
592:q
589:(
586:V
582:,
579:1
576:=
573:)
570:z
567:(
560:,
557:1
554:=
551:)
548:t
545:(
290:1
288:3
277:-
266:1
264:0
254:-
244:0
234:0
224:3
214:0
204:0
194:0
184:0
174:1
164:0
120:)
114:(
109:)
105:(
95:·
88:·
81:·
74:·
47:.
20:)
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