994:
24:
1006:
75:. In bridge representation, a knot lies entirely in the plane apart for a finite number of bridges whose projections onto the plane are straight lines. Equivalently, the bridge number is the minimal number of local maxima of the projection of the knot onto a vector, where we minimize over all projections and over all conformations of the knot. In this context, the bridge number is often called the
63:
Given a knot or link, draw a diagram of the link using the convention that a gap in the line denotes an undercrossing. Call an unbroken arc in this diagram a bridge if it includes at least one overcrossing. Then the bridge number of a knot can be found as the minimum number of bridges required for
371:
224:
939:
858:
343:
189:
89:
Every non-trivial knot has bridge number at least two, so the knots that minimize the bridge number (other than the
55:
of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot.
405:
212:
853:
848:
724:
137:
1010:
425:
487:
206:
175:
557:
552:
493:
364:
1032:
685:
899:
868:
216:
729:
179:
998:
769:
357:
317:
297:
234:
98:
8:
806:
789:
301:
827:
774:
388:
384:
287:
275:
924:
873:
823:
779:
739:
734:
652:
339:
220:
185:
152:
959:
784:
680:
415:
305:
257:
16:
This article is about a mathematical concept. For the telecommunications term, see
919:
883:
818:
764:
719:
712:
602:
514:
397:
313:
230:
120:, then the bridge number of K is one less than the sum of the bridge numbers of K
17:
349:
97:. It can be shown that every n-bridge knot can be decomposed into two trivial n-
979:
878:
840:
759:
672:
547:
539:
499:
142:
65:
52:
309:
1026:
914:
702:
695:
109:
102:
94:
929:
909:
813:
796:
592:
529:
147:
64:
any diagram of the knot. Bridge numbers were first studied in the 1950s by
36:
28:
248:
944:
707:
612:
481:
461:
451:
443:
435:
380:
72:
40:
964:
949:
904:
801:
754:
749:
744:
574:
471:
261:
71:
The bridge number can equivalently be defined geometrically instead of
969:
637:
292:
954:
564:
23:
215:, vol. 151, American Mathematical Society, Providence, RI,
974:
622:
582:
280:
Mathematical
Proceedings of the Cambridge Philosophical Society
90:
863:
934:
379:
278:(2003), "Additivity of bridge numbers of knots",
1024:
365:
184:, American Mathematical Society, p. 65,
131:
372:
358:
291:
274:
204:
247:
22:
170:
168:
1025:
353:
174:
1005:
165:
13:
328:
14:
1044:
1004:
993:
992:
213:Graduate Studies in Mathematics
859:Dowker–Thistlethwaite notation
268:
241:
198:
1:
158:
101:and hence 2-bridge knots are
84:
58:
205:Schultens, Jennifer (2014),
31:, drawn with bridge number 2
7:
208:Introduction to 3-manifolds
10:
1049:
132:Other numerical invariants
15:
988:
892:
849:Alexander–Briggs notation
836:
671:
573:
538:
396:
310:10.1017/S0305004103006832
250:Mathematische Zeitschrift
334:Cromwell, Peter (1994).
940:List of knots and links
488:Kinoshita–Terasaka knot
32:
730:Finite type invariant
26:
900:Alexander's theorem
302:2003MPCPS.135..539S
276:Schultens, Jennifer
262:10.1007/BF01181346
47:, also called the
33:
1020:
1019:
874:Reidemeister move
740:Khovanov homology
735:Hyperbolic volume
226:978-1-4704-1020-9
153:Unknotting number
1040:
1008:
1007:
996:
995:
960:Tait conjectures
663:
662:
648:
647:
633:
632:
525:
524:
510:
509:
494:(−2,3,7) pretzel
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367:
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322:
320:
295:
272:
266:
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245:
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202:
196:
194:
172:
1048:
1047:
1043:
1042:
1041:
1039:
1038:
1037:
1033:Knot invariants
1023:
1022:
1021:
1016:
984:
888:
854:Conway notation
838:
832:
819:Tricolorability
667:
661:
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656:
646:
643:
642:
641:
631:
628:
627:
626:
618:
608:
598:
588:
569:
548:Composite knots
534:
523:
520:
519:
518:
515:Borromean rings
508:
505:
504:
503:
477:
467:
457:
447:
439:
431:
421:
411:
392:
378:
336:Knots and Links
331:
329:Further reading
326:
325:
273:
269:
246:
242:
227:
203:
199:
192:
176:Adams, Colin C.
173:
166:
161:
138:Crossing number
134:
127:
123:
119:
115:
87:
61:
21:
18:Conference Call
12:
11:
5:
1046:
1036:
1035:
1018:
1017:
1015:
1014:
1002:
989:
986:
985:
983:
982:
980:Surgery theory
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972:
967:
962:
957:
952:
947:
942:
937:
932:
927:
922:
917:
912:
907:
902:
896:
894:
890:
889:
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886:
881:
879:Skein relation
876:
871:
866:
861:
856:
851:
845:
843:
834:
833:
831:
830:
824:Unknotting no.
821:
816:
811:
810:
809:
799:
794:
793:
792:
787:
782:
777:
772:
762:
757:
752:
747:
742:
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732:
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705:
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683:
677:
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669:
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659:
650:
644:
635:
629:
620:
616:
610:
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579:
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571:
570:
568:
567:
562:
561:
560:
555:
544:
542:
536:
535:
533:
532:
527:
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512:
506:
497:
491:
485:
479:
475:
469:
465:
459:
455:
449:
445:
441:
437:
433:
429:
423:
419:
413:
409:
402:
400:
394:
393:
377:
376:
369:
362:
354:
348:
347:
330:
327:
324:
323:
286:(3): 539–544,
267:
256:(1): 245–288.
240:
225:
197:
190:
163:
162:
160:
157:
156:
155:
150:
145:
143:Linking number
140:
133:
130:
125:
121:
117:
113:
103:rational knots
95:2-bridge knots
86:
83:
66:Horst Schubert
60:
57:
9:
6:
4:
3:
2:
1045:
1034:
1031:
1030:
1028:
1013:
1012:
1003:
1001:
1000:
991:
990:
987:
981:
978:
976:
973:
971:
968:
966:
963:
961:
958:
956:
953:
951:
948:
946:
943:
941:
938:
936:
933:
931:
928:
926:
923:
921:
918:
916:
915:Conway sphere
913:
911:
908:
906:
903:
901:
898:
897:
895:
891:
885:
882:
880:
877:
875:
872:
870:
867:
865:
862:
860:
857:
855:
852:
850:
847:
846:
844:
842:
835:
829:
825:
822:
820:
817:
815:
812:
808:
805:
804:
803:
800:
798:
795:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
767:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
714:
711:
710:
709:
706:
704:
701:
697:
694:
693:
692:
689:
687:
686:Arf invariant
684:
682:
679:
678:
676:
674:
670:
654:
651:
639:
636:
624:
621:
614:
611:
604:
601:
594:
591:
584:
581:
580:
578:
576:
572:
566:
563:
559:
556:
554:
551:
550:
549:
546:
545:
543:
541:
537:
531:
528:
516:
513:
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495:
492:
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486:
483:
480:
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442:
440:
434:
427:
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417:
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386:
382:
375:
370:
368:
363:
361:
356:
355:
352:
345:
344:9780521548311
341:
338:. Cambridge.
337:
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271:
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228:
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210:
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201:
193:
191:9780821886137
187:
183:
182:
181:The Knot Book
177:
171:
169:
164:
154:
151:
149:
146:
144:
141:
139:
136:
135:
129:
111:
110:connected sum
106:
104:
100:
96:
92:
82:
81:
78:
74:
73:topologically
69:
67:
56:
54:
50:
46:
45:bridge number
42:
38:
30:
25:
19:
1009:
997:
925:Double torus
910:Braid theory
725:Crossing no.
720:Crosscap no.
690:
406:Figure-eight
335:
293:math/0111032
283:
279:
270:
253:
249:
243:
207:
200:
180:
148:Stick number
108:If K is the
107:
88:
79:
76:
70:
62:
49:bridge index
48:
44:
37:mathematical
34:
29:trefoil knot
760:Linking no.
681:Alternating
482:Conway knot
462:Carrick mat
416:Three-twist
381:Knot theory
77:crookedness
41:knot theory
920:Complement
884:Tabulation
841:operations
765:Polynomial
755:Link group
750:Knot group
713:Invertible
691:Bridge no.
673:Invariants
603:Cinquefoil
472:Perko pair
398:Hyperbolic
159:References
93:) are the
85:Properties
59:Definition
814:Stick no.
770:Alexander
708:Chirality
653:Solomon's
613:Septafoil
540:Satellite
500:Whitehead
426:Stevedore
53:invariant
39:field of
1027:Category
999:Category
869:Mutation
837:Notation
790:Kauffman
703:Brunnian
696:2-bridge
565:Knot sum
496:(12n242)
178:(1994),
51:, is an
1011:Commons
930:Fibered
828:problem
797:Pretzel
775:Bracket
593:Trefoil
530:L10a140
490:(11n42)
484:(11n34)
452:Endless
318:2018265
298:Bibcode
235:3203728
99:tangles
35:In the
975:Writhe
945:Ribbon
780:HOMFLY
623:Unlink
583:Unknot
558:Square
553:Granny
342:
316:
233:
223:
217:p. 129
188:
91:unknot
43:, the
965:Twist
950:Slice
905:Berge
893:Other
864:Flype
802:Prime
785:Jones
745:Genus
575:Torus
389:links
385:knots
288:arXiv
124:and K
116:and K
970:Wild
935:Knot
839:and
826:and
807:list
638:Hopf
387:and
340:ISBN
221:ISBN
186:ISBN
112:of K
955:Sum
476:161
474:(10
306:doi
284:135
258:doi
1029::
655:(4
640:(2
625:(0
615:(7
605:(5
595:(3
585:(0
517:(6
502:(5
466:18
464:(8
454:(7
428:(6
418:(5
408:(4
314:MR
312:,
304:,
296:,
282:,
254:61
252:.
231:MR
229:,
219:,
211:,
167:^
128:.
105:.
68:.
27:A
664:)
660:1
649:)
645:1
634:)
630:1
619:)
617:1
609:)
607:1
599:)
597:1
589:)
587:1
526:)
522:2
511:)
507:1
478:)
468:)
458:)
456:4
446:3
444:6
438:2
436:6
432:)
430:1
422:)
420:2
412:)
410:1
391:)
383:(
373:e
366:t
359:v
346:.
321:.
308::
300::
290::
264:.
260::
238:.
195:.
126:2
122:1
118:2
114:1
80:.
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
