510:
772:
312:
396:
662:
810:
631:
869:
1265:
368:
196:
117:
1224:
1149:
1106:
1063:
547:
252:
78:
1201:
1172:
952:
590:
339:
1221:
1126:
1083:
1039:
1012:
992:
972:
929:
909:
889:
834:
567:
388:
272:
220:
161:
137:
670:
1284:
1174:
remains continuous. This behavior has many important consequences for energy considerations in many fields of science (Ricca 1997, 2005; Goriely 2006).
1296:
Banchoff, T.F. & White, J.H. (1975) The behavior of the total twist and self-linking number of a closed space curve under inversions.
1381:
41:
1316:
1268:
20:
505:{\displaystyle Tw={\dfrac {1}{2\pi }}\int \left(U\times {\dfrac {dU}{ds}}\right)\cdot {\dfrac {dX}{ds}}ds\;,}
1272:
277:
27:
639:
1342:, 411-429. Also in: (1995) Knots and Applications (ed. L.H. Kauffman), pp. 251-269. World Scientific.
1280:
36:
780:
1042:
813:
598:
839:
1386:
1276:
1229:
166:
87:
1131:
1088:
1048:
518:
225:
51:
1306:
Goriely, A. (2006) Twisted elastic rings and the rediscoveries of
Michell’s instability.
1320:
1183:
1154:
934:
572:
348:
321:
1358:
1345:
1332:
1223:, twist is a geometric quantity that plays an important role in the application of the
1206:
1111:
1068:
1024:
997:
977:
957:
914:
894:
874:
819:
767:{\displaystyle Tw={\dfrac {1}{2\pi }}\int \tau \;ds+{\dfrac {\left_{X}}{2\pi }}=T+N\;,}
552:
373:
342:
257:
205:
146:
122:
45:
1375:
1328:
199:
81:
140:
1348:(1997) Evolution and inflexional instability of twisted magnetic flux tubes.
1178:
1361:(2005) Inflexional disequilibrium of magnetic flux tubes.
1232:
1209:
1186:
1157:
1134:
1114:
1091:
1071:
1051:
1027:
1017:
When the ribbon is deformed so as to pass through an
1000:
980:
960:
937:
917:
897:
877:
842:
822:
783:
717:
684:
673:
642:
601:
575:
555:
521:
471:
441:
410:
399:
376:
351:
324:
280:
260:
228:
208:
169:
149:
125:
90:
54:
1322:
1128:simultaneously makes an equal and opposite jump of
1259:
1215:
1195:
1166:
1143:
1120:
1100:
1077:
1057:
1033:
1006:
986:
966:
946:
923:
903:
883:
863:
828:
804:
766:
656:
625:
592:can be decomposed (Moffatt & Ricca 1992) into
584:
561:
541:
504:
382:
362:
333:
306:
266:
246:
214:
190:
155:
131:
111:
72:
19:For twists of curves in algebraic geometry, see
1335:(1992) Helicity and the Calugareanu invariant.
390:. According to Love (1944) twist is defined by
8:
760:
706:
498:
1231:
1208:
1185:
1156:
1133:
1113:
1090:
1070:
1050:
1026:
999:
979:
959:
936:
916:
896:
876:
855:
841:
821:
782:
731:
716:
683:
672:
650:
649:
641:
600:
574:
554:
528:
520:
470:
440:
409:
398:
375:
350:
323:
279:
259:
227:
207:
168:
148:
124:
89:
53:
974:. Instead, only the normalized torsion
7:
1065:becomes singular. The total torsion
954:are independent of the ribbon field
871:denotes the total rotation angle of
848:
724:
307:{\displaystyle X'=X+\varepsilon U}
14:
657:{\displaystyle N\in \mathbb {Z} }
370:around and along the axial curve
202:, perpendicular at each point to
1225:Călugăreanu–White–Fuller formula
1151:(Moffatt & Ricca 1992) and
805:{\displaystyle \tau =\tau (s)}
799:
793:
620:
608:
549:is the unit tangent vector to
241:
229:
185:
179:
106:
100:
67:
55:
1:
1014:(Banchoff & White 1975).
994:is an invariant of the curve
1325:. Dover, 4th Ed., New York.
1271:(for its close relation to
1403:
1269:topological fluid dynamics
626:{\displaystyle T\in [0,1)}
18:
16:Differential geometry term
864:{\displaystyle \left_{X}}
569:. The total twist number
21:twists of elliptic curves
1260:{\displaystyle Lk=Wr+Tw}
594:normalized total torsion
1363:Fluid Dynamics Research
1337:Proc. R. Soc. London A
1261:
1217:
1197:
1168:
1145:
1122:
1102:
1079:
1059:
1035:
1008:
988:
968:
948:
925:
905:
885:
865:
830:
806:
768:
658:
627:
586:
563:
543:
506:
384:
364:
335:
308:
268:
248:
216:
192:
191:{\displaystyle U=U(s)}
157:
133:
113:
112:{\displaystyle X=X(s)}
74:
1382:Differential geometry
1285:structural complexity
1262:
1218:
1198:
1169:
1146:
1144:{\displaystyle \mp 1}
1123:
1103:
1101:{\displaystyle \pm 1}
1080:
1060:
1058:{\displaystyle \tau }
1036:
1009:
989:
969:
949:
926:
906:
886:
866:
831:
807:
769:
659:
628:
587:
564:
544:
542:{\displaystyle dX/ds}
507:
385:
365:
341:measures the average
336:
309:
269:
249:
247:{\displaystyle (X,U)}
217:
193:
158:
134:
114:
75:
73:{\displaystyle (X,U)}
28:differential geometry
1281:physical knot theory
1279:of a vector field),
1230:
1207:
1184:
1155:
1132:
1112:
1108:and the total angle
1089:
1069:
1049:
1025:
998:
978:
958:
935:
915:
895:
875:
840:
820:
781:
671:
640:
599:
573:
553:
519:
397:
374:
349:
322:
278:
258:
226:
206:
167:
147:
123:
88:
52:
1043:point of inflection
816:of the space curve
222:. Since the ribbon
1257:
1213:
1196:{\displaystyle Wr}
1193:
1177:Together with the
1167:{\displaystyle Tw}
1164:
1141:
1118:
1098:
1075:
1055:
1031:
1019:inflectional state
1004:
984:
964:
947:{\displaystyle Tw}
944:
921:
901:
881:
861:
826:
802:
764:
746:
698:
654:
623:
585:{\displaystyle Tw}
582:
559:
539:
502:
490:
460:
424:
380:
363:{\displaystyle X'}
360:
345:of the edge curve
334:{\displaystyle Tw}
331:
316:total twist number
304:
264:
244:
212:
188:
153:
129:
109:
70:
1277:magnetic helicity
1216:{\displaystyle X}
1121:{\displaystyle N}
1078:{\displaystyle T}
1034:{\displaystyle X}
1007:{\displaystyle X}
987:{\displaystyle T}
967:{\displaystyle U}
924:{\displaystyle N}
904:{\displaystyle X}
884:{\displaystyle U}
829:{\displaystyle X}
745:
697:
562:{\displaystyle X}
489:
459:
423:
383:{\displaystyle X}
267:{\displaystyle X}
215:{\displaystyle X}
156:{\displaystyle X}
132:{\displaystyle s}
80:be composed of a
1394:
1266:
1264:
1263:
1258:
1222:
1220:
1219:
1214:
1202:
1200:
1199:
1194:
1173:
1171:
1170:
1165:
1150:
1148:
1147:
1142:
1127:
1125:
1124:
1119:
1107:
1105:
1104:
1099:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1040:
1038:
1037:
1032:
1013:
1011:
1010:
1005:
993:
991:
990:
985:
973:
971:
970:
965:
953:
951:
950:
945:
930:
928:
927:
922:
910:
908:
907:
902:
890:
888:
887:
882:
870:
868:
867:
862:
860:
859:
854:
835:
833:
832:
827:
811:
809:
808:
803:
773:
771:
770:
765:
747:
744:
736:
735:
730:
718:
699:
696:
685:
663:
661:
660:
655:
653:
632:
630:
629:
624:
591:
589:
588:
583:
568:
566:
565:
560:
548:
546:
545:
540:
532:
511:
509:
508:
503:
491:
488:
480:
472:
466:
462:
461:
458:
450:
442:
425:
422:
411:
389:
387:
386:
381:
369:
367:
366:
361:
359:
340:
338:
337:
332:
314:, the twist (or
313:
311:
310:
305:
288:
273:
271:
270:
265:
253:
251:
250:
245:
221:
219:
218:
213:
197:
195:
194:
189:
162:
160:
159:
154:
138:
136:
135:
130:
118:
116:
115:
110:
79:
77:
76:
71:
1402:
1401:
1397:
1396:
1395:
1393:
1392:
1391:
1372:
1371:
1293:
1228:
1227:
1205:
1204:
1182:
1181:
1153:
1152:
1130:
1129:
1110:
1109:
1087:
1086:
1067:
1066:
1047:
1046:
1045:), the torsion
1023:
1022:
996:
995:
976:
975:
956:
955:
933:
932:
913:
912:
893:
892:
873:
872:
844:
843:
838:
837:
818:
817:
779:
778:
737:
720:
719:
689:
669:
668:
638:
637:
635:intrinsic twist
597:
596:
571:
570:
551:
550:
517:
516:
481:
473:
451:
443:
433:
429:
415:
395:
394:
372:
371:
352:
347:
346:
320:
319:
281:
276:
275:
256:
255:
224:
223:
204:
203:
165:
164:
145:
144:
121:
120:
86:
85:
50:
49:
48:. Let a ribbon
24:
17:
12:
11:
5:
1400:
1398:
1390:
1389:
1384:
1374:
1373:
1370:
1369:
1356:
1343:
1326:
1314:
1304:
1292:
1289:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1212:
1192:
1189:
1163:
1160:
1140:
1137:
1117:
1097:
1094:
1074:
1054:
1030:
1003:
983:
963:
943:
940:
920:
900:
880:
858:
853:
850:
847:
825:
801:
798:
795:
792:
789:
786:
775:
774:
763:
759:
756:
753:
750:
743:
740:
734:
729:
726:
723:
715:
712:
709:
705:
702:
695:
692:
688:
682:
679:
676:
652:
648:
645:
622:
619:
616:
613:
610:
607:
604:
581:
578:
558:
538:
535:
531:
527:
524:
513:
512:
501:
497:
494:
487:
484:
479:
476:
469:
465:
457:
454:
449:
446:
439:
436:
432:
428:
421:
418:
414:
408:
405:
402:
379:
358:
355:
330:
327:
303:
300:
297:
294:
291:
287:
284:
263:
243:
240:
237:
234:
231:
211:
187:
184:
181:
178:
175:
172:
152:
128:
108:
105:
102:
99:
96:
93:
69:
66:
63:
60:
57:
46:axial rotation
15:
13:
10:
9:
6:
4:
3:
2:
1399:
1388:
1385:
1383:
1380:
1379:
1377:
1367:
1364:
1360:
1357:
1354:
1351:
1350:Solar Physics
1347:
1344:
1341:
1338:
1334:
1330:
1329:Moffatt, H.K.
1327:
1324:
1323:
1318:
1315:
1312:
1309:
1305:
1302:
1299:
1295:
1294:
1290:
1288:
1286:
1282:
1278:
1274:
1270:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1226:
1210:
1190:
1187:
1180:
1175:
1161:
1158:
1138:
1135:
1115:
1095:
1092:
1072:
1052:
1044:
1028:
1020:
1015:
1001:
981:
961:
941:
938:
918:
898:
878:
856:
851:
845:
823:
815:
796:
790:
787:
784:
761:
757:
754:
751:
748:
741:
738:
732:
727:
721:
713:
710:
707:
703:
700:
693:
690:
686:
680:
677:
674:
667:
666:
665:
646:
643:
636:
617:
614:
611:
605:
602:
595:
579:
576:
556:
536:
533:
529:
525:
522:
499:
495:
492:
485:
482:
477:
474:
467:
463:
455:
452:
447:
444:
437:
434:
430:
426:
419:
416:
412:
406:
403:
400:
393:
392:
391:
377:
356:
353:
344:
328:
325:
317:
301:
298:
295:
292:
289:
285:
282:
261:
238:
235:
232:
209:
201:
200:normal vector
182:
176:
173:
170:
150:
142:
126:
103:
97:
94:
91:
83:
64:
61:
58:
47:
43:
39:
38:
33:
29:
22:
1365:
1362:
1352:
1349:
1339:
1336:
1321:
1317:Love, A.E.H.
1310:
1308:J Elasticity
1307:
1300:
1298:Math. Scand.
1297:
1176:
1018:
1016:
776:
634:
593:
514:
315:
35:
31:
25:
1359:Ricca, R.L.
1346:Ricca, R.L.
1333:Ricca, R.L.
911:. Neither
198:the a unit
82:space curve
1376:Categories
1368:, 319-332.
1355:, 241-248.
1313:, 281-299.
1303:, 254–262.
1291:References
1287:analysis.
254:has edges
141:arc length
1136:∓
1093:±
1085:jumps by
1053:τ
849:Θ
791:τ
785:τ
742:π
725:Θ
704:τ
701:∫
694:π
647:∈
606:∈
468:⋅
438:×
427:∫
420:π
299:ε
1387:Topology
357:′
286:′
119:, where
1319:(1944)
1273:kinetic
814:torsion
812:is the
343:winding
139:is the
40:is its
1331:&
1283:, and
1179:writhe
1041:has a
1021:(i.e.
891:along
836:, and
777:where
515:where
163:, and
37:ribbon
30:, the
34:of a
32:twist
1275:and
931:nor
633:and
274:and
42:rate
1353:172
1340:439
1267:in
1203:of
664:as
143:of
44:of
26:In
1378::
1366:36
1311:84
1301:36
318:)
84:,
1255:w
1252:T
1249:+
1246:r
1243:W
1240:=
1237:k
1234:L
1211:X
1191:r
1188:W
1162:w
1159:T
1139:1
1116:N
1096:1
1073:T
1029:X
1002:X
982:T
962:U
942:w
939:T
919:N
899:X
879:U
857:X
852:]
846:[
824:X
800:)
797:s
794:(
788:=
762:,
758:N
755:+
752:T
749:=
739:2
733:X
728:]
722:[
714:+
711:s
708:d
691:2
687:1
681:=
678:w
675:T
651:Z
644:N
621:)
618:1
615:,
612:0
609:[
603:T
580:w
577:T
557:X
537:s
534:d
530:/
526:X
523:d
500:,
496:s
493:d
486:s
483:d
478:X
475:d
464:)
456:s
453:d
448:U
445:d
435:U
431:(
417:2
413:1
407:=
404:w
401:T
378:X
354:X
329:w
326:T
302:U
296:+
293:X
290:=
283:X
262:X
242:)
239:U
236:,
233:X
230:(
210:X
186:)
183:s
180:(
177:U
174:=
171:U
151:X
127:s
107:)
104:s
101:(
98:X
95:=
92:X
68:)
65:U
62:,
59:X
56:(
23:.
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