771:
437:
390:
634:
Ribbon theory investigates geometric and topological aspects of a mathematical reference ribbon associated with physical and biological properties, such as those arising in
176:
457:
531:
269:
69:
205:
121:
748:
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225:
144:
89:
655:
724:
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92:
929:
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846:
627:
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17:
780:
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149:
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742:
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802:
242:
42:
954:
916:
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181:
97:
675:
784:
601:
568:
539:
561:
462:
346:
326:
306:
278:
210:
129:
74:
811:
968:
897:(1961), "Sur les classes d'isotopie des noeuds tridimensionels et leurs invariants",
670:
36:
927:
White, James H. (1969), "Self-linking and the Gauss integral in higher dimensions",
296:
763:
665:
32:
772:
Proceedings of the
National Academy of Sciences of the United States of America
734:
124:
29:
911:
843:
The Knot Book: An
Elementary Introduction to the Mathematical Theory of Knots
820:
793:
874:(1959), "L'intĂ©grale de Gauss et l'analyse des nĆuds tridimensionnels",
950:
594:
942:
227:
at each point. Ribbons have seen particular application as regards
639:
598:), a measure of non-planarity of the ribbon's axis curve; and
228:
565:, the integer number of turns of the ribbon around its axis;
631:), the rate of rotation of the ribbon around its axis.
604:
571:
542:
495:
465:
445:
398:
369:
349:
329:
309:
281:
245:
213:
184:
152:
132:
100:
77:
45:
613:
580:
551:
525:
482:The ribbon concept plays an important role in the
471:
451:
431:
384:
355:
335:
315:
287:
263:
219:
199:
170:
138:
115:
83:
63:
719:(First ed.). Boca Raton, FL. p. 49.
8:
363:. For any simple closed ribbon the curves
747:: CS1 maint: location missing publisher (
715:VologodskiÇ, Aleksandr Vadimovich (1992).
910:
876:Revue de Mathématiques Pure et Appliquées
810:
792:
603:
570:
541:
494:
464:
444:
439:are, for all sufficiently small positive
397:
368:
348:
328:
308:
280:
244:
212:
183:
151:
131:
99:
76:
44:
696:EinfĂŒhrung in die Differentialgeometrie
687:
764:"The writhing number of a space curve"
740:
299:(i.e. without self-intersections) and
123:, depending continuously on the curve
459:, simple closed curves disjoint from
432:{\displaystyle X(s)+\varepsilon U(s)}
39:. More formally, a ribbon denoted by
7:
717:Topology and Physics of Circular DNA
14:
899:Czechoslovak Mathematical Journal
323:and all its derivatives agree at
484:CÄlugÄreanu-White-Fuller formula
930:American Journal of Mathematics
385:{\displaystyle X+\varepsilon U}
426:
420:
408:
402:
258:
246:
194:
188:
110:
104:
58:
46:
1:
847:American Mathematical Society
171:{\displaystyle a\leq s\leq b}
91:given by a three-dimensional
452:{\displaystyle \varepsilon }
656:BollobĂĄsâRiordan polynomial
235:Properties and implications
996:
636:topological fluid dynamics
559:is the asymptotic (Gauss)
28:) is the combination of a
762:Fuller, F. Brock (1971).
526:{\displaystyle Lk=Wr+Tw,}
912:10.21136/CMJ.1961.100486
392:given parametrically by
615:
582:
553:
527:
473:
453:
433:
386:
357:
337:
317:
289:
265:
221:
201:
172:
140:
117:
85:
65:
35:and its corresponding
975:Differential geometry
895:CÄlugÄreanu, Gheorghe
872:CÄlugÄreanu, Gheorghe
794:10.1073/pnas.68.4.815
616:
583:
554:
528:
474:
454:
434:
387:
358:
338:
318:
290:
266:
264:{\displaystyle (X,U)}
222:
202:
178:), and a unit vector
173:
141:
118:
86:
66:
64:{\displaystyle (X,U)}
18:differential geometry
694:Blaschke, W. (1950)
602:
569:
540:
493:
463:
443:
396:
367:
347:
327:
307:
279:
243:
211:
200:{\displaystyle U(s)}
182:
150:
130:
116:{\displaystyle X(s)}
98:
75:
43:
785:1971PNAS...68..815B
698:. Springer-Verlag.
486:, that states that
614:{\displaystyle Tw}
611:
588:denotes the total
581:{\displaystyle Wr}
578:
552:{\displaystyle Lk}
549:
523:
469:
449:
429:
382:
353:
333:
313:
285:
261:
217:
197:
168:
136:
113:
81:
61:
472:{\displaystyle X}
356:{\displaystyle b}
336:{\displaystyle a}
316:{\displaystyle U}
288:{\displaystyle X}
220:{\displaystyle X}
207:perpendicular to
139:{\displaystyle s}
84:{\displaystyle X}
71:includes a curve
987:
961:
923:
914:
890:
867:
825:
824:
814:
796:
768:
759:
753:
752:
746:
738:
712:
706:
692:
661:Knots and graphs
644:material science
642:modeling and in
620:
618:
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612:
587:
585:
584:
579:
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430:
391:
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340:
339:
334:
322:
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270:
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206:
204:
203:
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177:
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169:
145:
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137:
122:
120:
119:
114:
90:
88:
87:
82:
70:
68:
67:
62:
995:
994:
990:
989:
988:
986:
985:
984:
965:
964:
943:10.2307/2373348
926:
893:
870:
857:
837:
834:
829:
828:
766:
761:
760:
756:
739:
727:
714:
713:
709:
693:
689:
684:
652:
600:
599:
590:writhing number
567:
566:
538:
537:
491:
490:
461:
460:
441:
440:
394:
393:
365:
364:
345:
344:
325:
324:
305:
304:
277:
276:
241:
240:
237:
209:
208:
180:
179:
148:
147:
128:
127:
96:
95:
73:
72:
41:
40:
12:
11:
5:
993:
991:
983:
982:
977:
967:
966:
963:
962:
937:(3): 693â728,
924:
891:
868:
855:
833:
830:
827:
826:
779:(4): 815â819.
754:
726:978-1138105058
725:
707:
686:
685:
683:
680:
679:
678:
673:
668:
663:
658:
651:
648:
610:
607:
577:
574:
562:linking number
548:
545:
534:
533:
522:
519:
516:
513:
510:
507:
504:
501:
498:
468:
448:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
381:
378:
375:
372:
352:
332:
312:
284:
260:
257:
254:
251:
248:
236:
233:
216:
196:
193:
190:
187:
167:
164:
161:
158:
155:
135:
112:
109:
106:
103:
80:
60:
57:
54:
51:
48:
13:
10:
9:
6:
4:
3:
2:
992:
981:
978:
976:
973:
972:
970:
960:
956:
952:
948:
944:
940:
936:
932:
931:
925:
922:
918:
913:
908:
904:
900:
896:
892:
889:
885:
881:
877:
873:
869:
866:
862:
858:
856:0-8218-3678-1
852:
848:
844:
840:
836:
835:
831:
822:
818:
813:
808:
804:
800:
795:
790:
786:
782:
778:
774:
773:
765:
758:
755:
750:
744:
736:
732:
728:
722:
718:
711:
708:
705:
704:9783817115495
701:
697:
691:
688:
681:
677:
674:
672:
671:DNA supercoil
669:
667:
664:
662:
659:
657:
654:
653:
649:
647:
645:
641:
637:
632:
630:
629:
624:
621:is the total
608:
605:
597:
596:
591:
575:
572:
564:
563:
546:
543:
520:
517:
514:
511:
508:
505:
502:
499:
496:
489:
488:
487:
485:
480:
466:
446:
423:
417:
414:
411:
405:
399:
379:
376:
373:
370:
350:
330:
310:
302:
298:
282:
274:
255:
252:
249:
234:
232:
230:
214:
191:
185:
165:
162:
159:
156:
153:
133:
126:
107:
101:
94:
78:
55:
52:
49:
38:
37:normal vector
34:
31:
27:
23:
19:
934:
928:
902:
898:
879:
875:
842:
839:Adams, Colin
832:Bibliography
776:
770:
757:
716:
710:
695:
690:
676:Möbius strip
633:
626:
623:twist number
622:
593:
589:
560:
535:
483:
481:
300:
297:simple curve
272:
238:
25:
21:
15:
905:: 588â625,
666:Knot theory
625:(or simply
592:(or simply
239:The ribbon
33:space curve
969:Categories
735:1014356603
682:References
271:is called
125:arc-length
743:cite book
447:ε
415:ε
377:ε
163:≤
157:≤
980:Topology
882:: 5â20,
841:(2004),
650:See also
959:0253264
951:2373348
921:0149378
888:0131846
865:2079925
821:5279522
803:0278197
781:Bibcode
303:and if
957:
949:
919:
886:
863:
853:
819:
812:389050
809:
801:
733:
723:
702:
595:writhe
536:where
301:closed
273:simple
93:vector
30:smooth
22:ribbon
947:JSTOR
767:(PDF)
628:twist
295:is a
26:strip
851:ISBN
817:PMID
749:link
731:OCLC
721:ISBN
700:ISBN
343:and
24:(or
20:, a
939:doi
907:doi
807:PMC
789:doi
640:DNA
275:if
229:DNA
16:In
971::
955:MR
953:,
945:,
935:91
933:,
917:MR
915:,
903:11
901:,
884:MR
878:,
861:MR
859:,
849:,
845:,
815:.
805:.
799:MR
797:.
787:.
777:68
775:.
769:.
745:}}
741:{{
729:.
646:.
638:,
479:.
231:.
941::
909::
880:4
823:.
791::
783::
751:)
737:.
609:w
606:T
576:r
573:W
547:k
544:L
521:,
518:w
515:T
512:+
509:r
506:W
503:=
500:k
497:L
467:X
427:)
424:s
421:(
418:U
412:+
409:)
406:s
403:(
400:X
380:U
374:+
371:X
351:b
331:a
311:U
283:X
259:)
256:U
253:,
250:X
247:(
215:X
195:)
192:s
189:(
186:U
166:b
160:s
154:a
146:(
134:s
111:)
108:s
105:(
102:X
79:X
59:)
56:U
53:,
50:X
47:(
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