47:
758:
The proof of this near-linear upper bound uses a divide-and-conquer argument to show that minimum projections of knots can be embedded as planar graphs in the cubic lattice. However, no one has yet observed a knot family with super-linear dependence of length on crossing number and it is conjectured
756:
359:. However, the search for the answer has spurred research on both theoretical and computational ground. It has been shown that for each link type there is a ropelength minimizer although it may only be of
661:
145:
665:
On the other hand, there also exist knots whose ropelength is larger, proportional to the crossing number itself rather than to a smaller power of it. This is nearly tight, as for every knot,
486:
177:
518:
226:
668:
556:
35:. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are called
387:
357:
333:
579:
439:
419:
297:
277:
250:
197:
81:
829:
Gonzalez, O.; Maddocks, J. H.; Schuricht, F.; von der Mosel, H. (2002), "Global curvature and self-contact of nonlinearly elastic curves and rods",
999:
Diao, Yuanan; Ernst, Claus; Por, Attila; Ziegler, Uta (2019), "The
Ropelengths of Knots Are Almost Linear in Terms of Their Crossing Numbers",
592:
389:. For the simplest nontrivial knot, the trefoil knot, computer simulations have shown that its minimum ropelength is at most 16.372.
867:
780:
1026:
86:
398:
1077:
397:
An extensive search has been devoted to showing relations between ropelength and other knot invariants such as the
444:
1072:
965:
Diao, Y.; Ernst, C.; Thistlethwaite, M. (2003), "The linear growth in the lengths of a family of thick knots",
879:
150:
491:
785:
898:
20:
888:
794:
28:
24:
202:
523:
1008:
974:
940:
906:
838:
804:
1045:
986:
952:
918:
850:
816:
365:
342:
1041:
982:
948:
914:
871:
846:
812:
902:
318:
586:
564:
424:
404:
282:
262:
256:
235:
229:
182:
66:
944:
1066:
1049:
336:
51:
931:
Buck, Gregory; Simon, Jonathan (1999), "Thickness and crossing number of knots",
751:{\displaystyle L(K)=O(\operatorname {Cr} (K)\log ^{5}(\operatorname {Cr} (K))).}
1012:
978:
910:
558:
360:
783:(2006), "Quadrisecants give new lower bounds for the ropelength of a knot",
582:
307:
One of the earliest knot theory questions was posed in the following terms:
46:
842:
808:
893:
799:
520:
denotes the crossing number. There exist knots and links, namely the
315:
In terms of ropelength, this asks if there is a knot with ropelength
585:, for which this lower bound is tight. That is, for these knots (in
339:
shows that the ropelength of any nontrivial knot has to be at least
45:
1027:"Realizable powers of ropelengths by non-trivial knot families"
311:
Can I tie a knot on a foot-long rope that is one inch thick?
279:
to be the minimum ropelength over all curves that realize
831:
Calculus of
Variations and Partial Differential Equations
671:
656:{\displaystyle L(K)=O(\operatorname {Cr} (K)^{3/4}).}
595:
567:
526:
494:
447:
427:
407:
368:
345:
321:
285:
265:
238:
205:
185:
153:
140:{\displaystyle L(C)=\operatorname {Len} (C)/\tau (C)}
89:
69:
750:
655:
573:
550:
512:
480:
433:
413:
381:
351:
327:
291:
271:
244:
220:
191:
171:
139:
75:
872:"On the minimum ropelength of knots and links"
759:that the tight upper bound should be linear.
8:
1001:Journal of Knot Theory and its Ramifications
967:Journal of Knot Theory and its Ramifications
481:{\displaystyle \operatorname {Cr} (K)^{3/4}}
892:
798:
712:
670:
637:
633:
594:
566:
525:
493:
468:
464:
446:
426:
406:
373:
367:
344:
320:
284:
264:
237:
204:
184:
152:
120:
88:
68:
768:
335:. The answer is no: an argument using
172:{\displaystyle \operatorname {Len} (C)}
866:Cantarella, Jason; Kusner, Robert B.;
513:{\displaystyle \operatorname {Cr} (K)}
861:
859:
774:
772:
259:by defining the ropelength of a knot
7:
50:A numeric approximation of an ideal
1034:JP Journal of Geometry and Topology
1025:Diao, Yuanan; Ernst, Claus (2004),
63:The ropelength of a knotted curve
14:
779:Denne, Elizabeth; Diao, Yuanan;
255:Ropelength can be turned into a
742:
739:
736:
730:
721:
705:
699:
690:
681:
675:
647:
630:
623:
614:
605:
599:
545:
527:
507:
501:
461:
454:
215:
209:
166:
160:
134:
128:
117:
111:
99:
93:
1:
945:10.1016/S0166-8641(97)00211-3
933:Topology and its Applications
393:Dependence on crossing number
441:is at least proportional to
1094:
401:of a knot. For every knot
1013:10.1142/S0218216519500858
979:10.1142/S0218216503002615
911:10.1007/s00222-002-0234-y
880:Inventiones Mathematicae
221:{\displaystyle \tau (C)}
83:is defined as the ratio
23:, each realization of a
786:Geometry & Topology
551:{\displaystyle (k,k-1)}
361:differentiability class
752:
657:
575:
552:
514:
482:
435:
415:
383:
353:
329:
293:
273:
246:
222:
193:
173:
141:
77:
55:
843:10.1007/s005260100089
753:
658:
576:
553:
515:
483:
436:
416:
384:
382:{\displaystyle C^{1}}
354:
352:{\displaystyle 15.66}
330:
303:Ropelength minimizers
294:
274:
247:
223:
194:
174:
142:
78:
49:
809:10.2140/gt.2006.10.1
669:
593:
565:
524:
492:
445:
425:
421:, the ropelength of
405:
366:
343:
319:
283:
263:
236:
203:
183:
151:
87:
67:
21:physical knot theory
903:2002InMat.150..257C
1078:Geometric topology
748:
653:
571:
548:
510:
478:
431:
411:
379:
349:
328:{\displaystyle 12}
325:
289:
269:
242:
218:
189:
169:
137:
73:
56:
31:has an associated
868:Sullivan, John M.
781:Sullivan, John M.
574:{\displaystyle k}
434:{\displaystyle K}
414:{\displaystyle K}
292:{\displaystyle K}
272:{\displaystyle K}
245:{\displaystyle C}
192:{\displaystyle C}
179:is the length of
76:{\displaystyle C}
1085:
1057:
1056:
1054:
1048:, archived from
1031:
1022:
1016:
1015:
996:
990:
989:
962:
956:
955:
928:
922:
921:
896:
876:
863:
854:
853:
826:
820:
819:
802:
776:
757:
755:
754:
749:
717:
716:
662:
660:
659:
654:
646:
645:
641:
580:
578:
577:
572:
557:
555:
554:
549:
519:
517:
516:
511:
487:
485:
484:
479:
477:
476:
472:
440:
438:
437:
432:
420:
418:
417:
412:
388:
386:
385:
380:
378:
377:
358:
356:
355:
350:
334:
332:
331:
326:
298:
296:
295:
290:
278:
276:
275:
270:
251:
249:
248:
243:
227:
225:
224:
219:
198:
196:
195:
190:
178:
176:
175:
170:
146:
144:
143:
138:
124:
82:
80:
79:
74:
1093:
1092:
1088:
1087:
1086:
1084:
1083:
1082:
1073:Knot invariants
1063:
1062:
1061:
1060:
1052:
1029:
1024:
1023:
1019:
1007:(14): 1950085,
998:
997:
993:
964:
963:
959:
930:
929:
925:
874:
865:
864:
857:
828:
827:
823:
778:
777:
770:
765:
708:
667:
666:
629:
591:
590:
563:
562:
522:
521:
490:
489:
460:
443:
442:
423:
422:
403:
402:
399:crossing number
395:
369:
364:
363:
341:
340:
317:
316:
313:
305:
281:
280:
261:
260:
234:
233:
201:
200:
181:
180:
149:
148:
85:
84:
65:
64:
61:
17:
12:
11:
5:
1091:
1089:
1081:
1080:
1075:
1065:
1064:
1059:
1058:
1040:(2): 197–208,
1017:
991:
973:(5): 709–715,
957:
939:(3): 245–257,
923:
887:(2): 257–286,
855:
821:
767:
766:
764:
761:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
715:
711:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
652:
649:
644:
640:
636:
632:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
587:big O notation
570:
547:
544:
541:
538:
535:
532:
529:
509:
506:
503:
500:
497:
475:
471:
467:
463:
459:
456:
453:
450:
430:
410:
394:
391:
376:
372:
348:
324:
309:
304:
301:
288:
268:
257:knot invariant
241:
230:knot thickness
217:
214:
211:
208:
188:
168:
165:
162:
159:
156:
136:
133:
130:
127:
123:
119:
116:
113:
110:
107:
104:
101:
98:
95:
92:
72:
60:
57:
43:respectively.
16:Knot invariant
15:
13:
10:
9:
6:
4:
3:
2:
1090:
1079:
1076:
1074:
1071:
1070:
1068:
1055:on 2005-02-15
1051:
1047:
1043:
1039:
1035:
1028:
1021:
1018:
1014:
1010:
1006:
1002:
995:
992:
988:
984:
980:
976:
972:
968:
961:
958:
954:
950:
946:
942:
938:
934:
927:
924:
920:
916:
912:
908:
904:
900:
895:
890:
886:
882:
881:
873:
869:
862:
860:
856:
852:
848:
844:
840:
836:
832:
825:
822:
818:
814:
810:
806:
801:
796:
792:
788:
787:
782:
775:
773:
769:
762:
760:
745:
733:
727:
724:
718:
713:
709:
702:
696:
693:
687:
684:
678:
672:
663:
650:
642:
638:
634:
626:
620:
617:
611:
608:
602:
596:
588:
584:
568:
560:
542:
539:
536:
533:
530:
504:
498:
495:
473:
469:
465:
457:
451:
448:
428:
408:
400:
392:
390:
374:
370:
362:
346:
338:
337:quadrisecants
322:
312:
308:
302:
300:
286:
266:
258:
253:
239:
231:
212:
206:
186:
163:
157:
154:
131:
125:
121:
114:
108:
105:
102:
96:
90:
70:
58:
53:
48:
44:
42:
38:
34:
30:
26:
22:
1050:the original
1037:
1033:
1020:
1004:
1000:
994:
970:
966:
960:
936:
932:
926:
894:math/0103224
884:
878:
837:(1): 29–68,
834:
830:
824:
800:math/0408026
790:
784:
664:
396:
314:
310:
306:
254:
62:
40:
36:
32:
18:
559:torus knots
41:ideal links
37:ideal knots
1067:Categories
763:References
583:Hopf links
59:Definition
33:ropelength
728:
719:
697:
621:
540:−
499:
452:
207:τ
158:
126:τ
109:
870:(2002),
793:: 1–26,
488:, where
147:, where
1046:2105812
987:1999639
953:1666650
919:1933586
899:Bibcode
851:1883599
817:2207788
228:is the
52:trefoil
1044:
985:
951:
917:
849:
815:
1053:(PDF)
1030:(PDF)
889:arXiv
875:(PDF)
795:arXiv
347:15.66
561:and
199:and
39:and
29:knot
25:link
1009:doi
975:doi
941:doi
907:doi
885:150
839:doi
805:doi
710:log
589:),
232:of
155:Len
106:Len
27:or
19:In
1069::
1042:MR
1036:,
1032:,
1005:28
1003:,
983:MR
981:,
971:12
969:,
949:MR
947:,
937:91
935:,
915:MR
913:,
905:,
897:,
883:,
877:,
858:^
847:MR
845:,
835:14
833:,
813:MR
811:,
803:,
791:10
789:,
771:^
725:Cr
694:Cr
618:Cr
496:Cr
449:Cr
323:12
299:.
252:.
1038:4
1011::
977::
943::
909::
901::
891::
841::
807::
797::
746:.
743:)
740:)
737:)
734:K
731:(
722:(
714:5
706:)
703:K
700:(
691:(
688:O
685:=
682:)
679:K
676:(
673:L
651:.
648:)
643:4
639:/
635:3
631:)
627:K
624:(
615:(
612:O
609:=
606:)
603:K
600:(
597:L
581:-
569:k
546:)
543:1
537:k
534:,
531:k
528:(
508:)
505:K
502:(
474:4
470:/
466:3
462:)
458:K
455:(
429:K
409:K
375:1
371:C
287:K
267:K
240:C
216:)
213:C
210:(
187:C
167:)
164:C
161:(
135:)
132:C
129:(
122:/
118:)
115:C
112:(
103:=
100:)
97:C
94:(
91:L
71:C
54:.
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