31:
966:
978:
90:
in order of traversal (each crossing is visited and labelled twice), with the following modification: if the label is an even number and the strand followed crosses over at the crossing, then change the sign on the label to be a negative. When finished, each crossing will be labelled a pair of
107:
may have crossings labelled with the pairs (1, 6) (3, −12) (5, 2) (7, 8) (9, −4) and (11, −10). The Dowker–Thistlethwaite notation for this labelling is the sequence: 6 −12 2 8 −4 −10.
127:
In the more general case, a knot can be recovered from a Dowker–Thistlethwaite sequence, but the recovered knot may differ from the original by either being a reflection or by having any
131:
component reflected in the line between its entry/exit points – the Dowker–Thistlethwaite notation is unchanged by these reflections. Knots tabulations typically consider only
86:
To generate the Dowker–Thistlethwaite notation, traverse the knot using an arbitrary starting point and direction. Label each of the n crossings with the numbers 1, ..., 2
91:
integers, one even and one odd. The Dowker–Thistlethwaite notation is the sequence of even integer labels associated with the labels 1, 3, ..., 2
343:
278:
911:
377:
17:
1009:
825:
820:
696:
160:
155:
982:
397:
459:
146:, posed by Tait, concerns counting the number of different number sequences possible in this notation.
529:
524:
465:
336:
307:
657:
316:
871:
840:
701:
71:
1004:
970:
741:
329:
67:
8:
778:
761:
799:
746:
360:
356:
75:
59:
295:
896:
845:
795:
751:
711:
706:
624:
274:
211:
206:
189:
143:
931:
756:
652:
387:
245:
201:
891:
855:
790:
736:
691:
684:
574:
486:
369:
321:
951:
850:
812:
731:
644:
519:
511:
471:
300:
250:
233:
165:
998:
886:
674:
667:
662:
271:
The Knot Book: An
Elementary Introduction to the Mathematical Theory of Knots
215:
128:
901:
881:
785:
768:
564:
501:
229:
104:
30:
916:
679:
584:
453:
433:
423:
415:
407:
352:
43:
39:
936:
921:
876:
773:
726:
721:
716:
546:
443:
241:
132:
117:
941:
609:
136:
926:
536:
116:
63:
946:
594:
554:
835:
121:
232:; Halverson, James; Ruehle, Fabian; Sułkowsk, Piotr (2021).
228:
34:
A knot diagram with crossings labelled for a Dowker sequence
906:
27:
Mathematical notation for describing the structure of knots
188:
Dowker, C. H.; Thistlethwaite, Morwen B. (1983-07-01).
187:
139:, so this ambiguity does not affect the tabulation.
351:
996:
317:What are Gauss and Dowker-Thistlethwaite codes?
273:. Providence, R.I.: American Mathematical Soc.
337:
74:, who refined a notation originally due to
344:
330:
111:
249:
205:
238:Machine Learning: Science and Technology
183:
181:
29:
14:
997:
325:
268:
178:
977:
190:"Classification of knot projections"
24:
262:
25:
1021:
288:
976:
965:
964:
296:DT (Dowker-Thistlethwaite) Codes
66:. The notation is named after
831:Dowker–Thistlethwaite notation
222:
95: − 1 in turn.
13:
1:
194:Topology and Its Applications
171:
81:
269:Adams, Colin Conrad (2001).
207:10.1016/0166-8641(83)90004-4
7:
149:
48:Dowker–Thistlethwaite
10:
1026:
98:
960:
864:
821:Alexander–Briggs notation
808:
643:
545:
510:
368:
156:Alexander–Briggs notation
251:10.1088/2632-2153/abe91f
912:List of knots and links
460:Kinoshita–Terasaka knot
112:Uniqueness and counting
62:is a sequence of even
35:
1010:Mathematical notation
702:Finite type invariant
72:Morwen Thistlethwaite
33:
234:"Learning to unknot"
68:Clifford Hugh Dowker
872:Alexander's theorem
76:Peter Guthrie Tait
36:
992:
991:
846:Reidemeister move
712:Khovanov homology
707:Hyperbolic volume
280:978-0-8218-3678-1
16:(Redirected from
1017:
980:
979:
968:
967:
932:Tait conjectures
635:
634:
620:
619:
605:
604:
497:
496:
482:
481:
466:(−2,3,7) pretzel
346:
339:
332:
323:
322:
284:
256:
255:
253:
226:
220:
219:
209:
185:
21:
1025:
1024:
1020:
1019:
1018:
1016:
1015:
1014:
995:
994:
993:
988:
956:
860:
826:Conway notation
810:
804:
791:Tricolorability
639:
633:
630:
629:
628:
618:
615:
614:
613:
603:
600:
599:
598:
590:
580:
570:
560:
541:
520:Composite knots
506:
495:
492:
491:
490:
487:Borromean rings
480:
477:
476:
475:
449:
439:
429:
419:
411:
403:
393:
383:
364:
350:
291:
281:
265:
263:Further reading
260:
259:
227:
223:
186:
179:
174:
161:Conway notation
152:
114:
103:For example, a
101:
84:
58:or code, for a
28:
23:
22:
18:Dowker notation
15:
12:
11:
5:
1023:
1013:
1012:
1007:
990:
989:
987:
986:
974:
961:
958:
957:
955:
954:
952:Surgery theory
949:
944:
939:
934:
929:
924:
919:
914:
909:
904:
899:
894:
889:
884:
879:
874:
868:
866:
862:
861:
859:
858:
853:
851:Skein relation
848:
843:
838:
833:
828:
823:
817:
815:
806:
805:
803:
802:
796:Unknotting no.
793:
788:
783:
782:
781:
771:
766:
765:
764:
759:
754:
749:
744:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
688:
687:
677:
672:
671:
670:
660:
655:
649:
647:
641:
640:
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637:
631:
622:
616:
607:
601:
592:
588:
582:
578:
572:
568:
562:
558:
551:
549:
543:
542:
540:
539:
534:
533:
532:
527:
516:
514:
508:
507:
505:
504:
499:
493:
484:
478:
469:
463:
457:
451:
447:
441:
437:
431:
427:
421:
417:
413:
409:
405:
401:
395:
391:
385:
381:
374:
372:
366:
365:
349:
348:
341:
334:
326:
320:
319:
314:
305:
301:The Knot Atlas
290:
289:External links
287:
286:
285:
279:
264:
261:
258:
257:
221:
176:
175:
173:
170:
169:
168:
166:Gauss notation
163:
158:
151:
148:
144:ménage problem
135:and disregard
113:
110:
100:
97:
83:
80:
26:
9:
6:
4:
3:
2:
1022:
1011:
1008:
1006:
1003:
1002:
1000:
985:
984:
975:
973:
972:
963:
962:
959:
953:
950:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
908:
905:
903:
900:
898:
895:
893:
890:
888:
887:Conway sphere
885:
883:
880:
878:
875:
873:
870:
869:
867:
863:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
818:
816:
814:
807:
801:
797:
794:
792:
789:
787:
784:
780:
777:
776:
775:
772:
770:
767:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
739:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
686:
683:
682:
681:
678:
676:
673:
669:
666:
665:
664:
661:
659:
658:Arf invariant
656:
654:
651:
650:
648:
646:
642:
626:
623:
611:
608:
596:
593:
586:
583:
576:
573:
566:
563:
556:
553:
552:
550:
548:
544:
538:
535:
531:
528:
526:
523:
522:
521:
518:
517:
515:
513:
509:
503:
500:
488:
485:
473:
470:
467:
464:
461:
458:
455:
452:
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442:
435:
432:
425:
422:
420:
414:
412:
406:
399:
396:
389:
386:
379:
376:
375:
373:
371:
367:
362:
358:
354:
347:
342:
340:
335:
333:
328:
327:
324:
318:
315:
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306:
303:
302:
297:
293:
292:
282:
276:
272:
267:
266:
252:
247:
243:
239:
235:
231:
230:Gukov, Sergei
225:
217:
213:
208:
203:
199:
195:
191:
184:
182:
177:
167:
164:
162:
159:
157:
154:
153:
147:
145:
140:
138:
134:
130:
129:connected sum
125:
123:
119:
109:
106:
96:
94:
89:
79:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
32:
19:
981:
969:
897:Double torus
882:Braid theory
830:
697:Crossing no.
692:Crosscap no.
378:Figure-eight
311:
299:
270:
237:
224:
200:(1): 19–31.
197:
193:
141:
126:
124:reflection.
115:
105:knot diagram
102:
92:
87:
85:
55:
51:
47:
40:mathematical
37:
1005:Knot theory
732:Linking no.
653:Alternating
454:Conway knot
434:Carrick mat
388:Three-twist
353:Knot theory
308:DT Notation
133:prime knots
118:prime knots
44:knot theory
999:Categories
892:Complement
856:Tabulation
813:operations
737:Polynomial
727:Link group
722:Knot group
685:Invertible
663:Bridge no.
645:Invariants
575:Cinquefoil
444:Perko pair
370:Hyperbolic
242:IOPscience
172:References
120:uniquely,
82:Definition
786:Stick no.
742:Alexander
680:Chirality
625:Solomon's
585:Septafoil
512:Satellite
472:Whitehead
398:Stevedore
216:0166-8641
137:chirality
42:field of
971:Category
841:Mutation
809:Notation
762:Kauffman
675:Brunnian
668:2-bridge
537:Knot sum
468:(12n242)
312:Knotinfo
150:See also
64:integers
56:notation
983:Commons
902:Fibered
800:problem
769:Pretzel
747:Bracket
565:Trefoil
502:L10a140
462:(11n42)
456:(11n34)
424:Endless
99:Example
38:In the
947:Writhe
917:Ribbon
752:HOMFLY
595:Unlink
555:Unknot
530:Square
525:Granny
277:
214:
46:, the
937:Twist
922:Slice
877:Berge
865:Other
836:Flype
774:Prime
757:Jones
717:Genus
547:Torus
361:links
357:knots
122:up to
942:Wild
907:Knot
811:and
798:and
779:list
610:Hopf
359:and
275:ISBN
212:ISSN
142:The
70:and
60:knot
927:Sum
448:161
446:(10
298:",
246:doi
202:doi
1001::
627:(4
612:(2
597:(0
587:(7
577:(5
567:(3
557:(0
489:(6
474:(5
438:18
436:(8
426:(7
400:(6
390:(5
380:(4
310:,
244:.
240:.
236:.
210:.
198:16
196:.
192:.
180:^
78:.
54:)
52:DT
636:)
632:1
621:)
617:1
606:)
602:1
591:)
589:1
581:)
579:1
571:)
569:1
561:)
559:1
498:)
494:2
483:)
479:1
450:)
440:)
430:)
428:4
418:3
416:6
410:2
408:6
404:)
402:1
394:)
392:2
384:)
382:1
363:)
355:(
345:e
338:t
331:v
304:.
294:"
283:.
254:.
248::
218:.
204::
93:n
88:n
50:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
