1218:
1230:
17:
96:
In
November 2015, Joshua Evan Greene published a preprint that established a characterization of alternating links in terms of definite spanning surfaces, i.e. a definition of alternating links (of which alternating knots are a special case) without using the concept of a
128:
135:
123:. Each crossing is associated with an edge and half of the connected components of the complement of the diagram are associated with vertices in a checker board manner.
93:
to ask, "What is an alternating knot?" By this he was asking what non-diagrammatic properties of the knot complement would characterize alternating knots.
109:
59:, was what enabled early knot tabulators, such as Tait, to construct tables with relatively few mistakes or omissions. The simplest non-alternating
199:
78:
It is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly.
595:
437:
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1163:
1082:
405:
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629:
44:
if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is
112:, alternating diagram is the crossing number of the knot. This last is one of the celebrated Tait conjectures.
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948:
52:
21:
1234:
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423:
781:
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717:
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104:
Various geometric and topological information is revealed in an alternating diagram. Primeness and
909:
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55:
less than 10 are alternating. This fact and useful properties of alternating knots, such as the
1123:
1092:
953:
305:
203:
191:
1222:
993:
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277:, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, pp. 32–40,
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has shown that the volume has upper and lower linear bounds as functions of the number of
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37:
33:
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876:
550:
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531:
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433:
401:
363:
286:
514:
348:
1183:
1008:
639:
502:
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336:
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222:
146:
56:
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86:
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230:
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926:
919:
914:
480:
429:
The Knot Book: An
Elementary Introduction to the Mathematical Theory of Knots
249:
155:
Any reduced diagram of an alternating link has the fewest possible crossings.
89:
having useful and interesting geometric and topological properties. This led
400:. Annals of Mathematics Studies. Vol. 115. Princeton University Press.
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452:"Closed incompressible surfaces in alternating knot and link complements"
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978:
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242:
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60:
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371:
90:
81:
Alternating links end up having an important role in knot theory and
548:
361:
158:
Any two reduced diagrams of the same alternating knot have the same
108:
of a link is easily seen from the diagram. The crossing number of a
1178:
788:
331:
317:
Greene, Joshua (2017). "Alternating links and definite surfaces".
273:
Lickorish, W. B. Raymond (1997), "Geometry of
Alternating Links",
248:
Thus hyperbolic volume is an invariant of many alternating links.
483:(2004). "The volume of hyperbolic alternating link complements".
1198:
846:
806:
159:
1087:
183:
16:
1158:
567:
529:
127:
134:
233:, showed that any prime, non-split alternating link is
181:
by means of a sequence of certain simple moves called
570:to build an alternating knot from its planar graph
202:proved the first two Tait conjectures in 1987 and
603:
1248:
589:
63:have 8 crossings (and there are three such: 8
210:proved the Tait flyping conjecture in 1991.
187:. Also known as the Tait flyping conjecture.
165:Given any two reduced alternating diagrams D
596:
582:
496:
470:
330:
272:
173:of an oriented, prime alternating link: D
479:
392:
20:One of three non-alternating knots with
15:
446:
119:is in one-to-one correspondence with a
1249:
316:
577:
549:
530:
422:
362:
1229:
213:
256:of a reduced, alternating diagram.
140:
13:
385:
48:if it has an alternating diagram.
14:
1273:
523:
432:. American Mathematical Society.
237:, i.e. the link complement has a
1228:
1217:
1216:
133:
126:
1083:Dowker–Thistlethwaite notation
355:
310:
275:An Introduction to Knot Theory
266:
1:
259:
472:10.1016/0040-9383(84)90023-5
7:
1257:Alternating knots and links
283:10.1007/978-1-4612-0691-0_4
10:
1278:
341:10.1215/00127094-2017-0004
151:The Tait conjectures are:
144:
1212:
1116:
1073:Alexander–Briggs notation
1060:
895:
797:
762:
620:
554:"Tait's Knot Conjectures"
507:10.1112/S0024611503014291
367:"Tait's Knot Conjectures"
319:Duke Mathematical Journal
1164:List of knots and links
712:Kinoshita–Terasaka knot
241:, unless the link is a
227:hyperbolization theorem
177:may be transformed to D
51:Many of the knots with
485:Proc. London Math. Soc
378:Accessed: May 5, 2013.
25:
954:Finite type invariant
204:Morwen Thistlethwaite
192:Morwen Thistlethwaite
85:theory, due to their
19:
304:; see in particular
1124:Alexander's theorem
239:hyperbolic geometry
551:Weisstein, Eric W.
535:"Alternating Knot"
532:Weisstein, Eric W.
394:Kauffman, Louis H.
364:Weisstein, Eric W.
26:
1244:
1243:
1098:Reidemeister move
964:Khovanov homology
959:Hyperbolic volume
439:978-0-8218-3678-1
214:Hyperbolic volume
1269:
1232:
1231:
1220:
1219:
1184:Tait conjectures
887:
886:
872:
871:
857:
856:
749:
748:
734:
733:
718:(−2,3,7) pretzel
598:
591:
584:
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545:
544:
518:
500:
476:
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448:Menasco, William
443:
419:
379:
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359:
353:
352:
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147:Tait conjectures
141:Tait conjectures
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57:Tait conjectures
1277:
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1268:
1267:
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1262:Knot invariants
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1246:
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1112:
1078:Conway notation
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1043:Tricolorability
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772:Composite knots
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739:Borromean rings
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568:Celtic Knotwork
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424:Adams, Colin C.
408:
388:
386:Further reading
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293:
271:
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231:Haken manifolds
216:
208:William Menasco
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115:An alternating
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70:
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53:crossing number
22:crossing number
12:
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5:
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1204:Surgery theory
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1103:Skein relation
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1095:
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1080:
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1069:
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1054:
1048:Unknotting no.
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736:
730:
721:
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709:
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673:
669:
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524:External links
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519:
491:(1): 204–224.
481:Lackenby, Marc
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196:Louis Kauffman
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145:Main article:
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1139:Conway sphere
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910:Arf invariant
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407:0-691-08435-1
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292:0-387-98254-X
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257:
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254:twist regions
251:
250:Marc Lackenby
246:
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236:
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107:
106:splittability
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100:
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62:
58:
54:
49:
47:
43:
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35:
31:
23:
18:
1233:
1221:
1149:Double torus
1134:Braid theory
949:Crossing no.
944:Crosscap no.
904:
630:Figure-eight
557:
538:
498:math/0012185
488:
484:
465:(1): 37–44.
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458:
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357:
322:
318:
312:
274:
268:
253:
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217:
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125:
121:planar graph
117:knot diagram
114:
103:
99:link diagram
95:
80:
77:
50:
45:
41:
27:
984:Linking no.
905:Alternating
706:Conway knot
686:Carrick mat
640:Three-twist
605:Knot theory
221:, applying
200:K. Murasugi
87:complements
61:prime knots
46:alternating
42:alternating
40:diagram is
30:knot theory
1251:Categories
1144:Complement
1108:Tabulation
1065:operations
989:Polynomial
979:Link group
974:Knot group
937:Invertible
915:Bridge no.
897:Invariants
827:Cinquefoil
696:Perko pair
622:Hyperbolic
416:0627.57002
332:1511.06329
260:References
243:torus link
235:hyperbolic
83:3-manifold
1038:Stick no.
994:Alexander
932:Chirality
877:Solomon's
837:Septafoil
764:Satellite
724:Whitehead
650:Stevedore
559:MathWorld
540:MathWorld
372:MathWorld
91:Ralph Fox
1223:Category
1093:Mutation
1061:Notation
1014:Kauffman
927:Brunnian
920:2-bridge
789:Knot sum
720:(12n242)
515:56284382
459:Topology
450:(1984).
426:(2004).
398:On Knots
396:(1987).
349:59023367
223:Thurston
1235:Commons
1154:Fibered
1052:problem
1021:Pretzel
999:Bracket
817:Trefoil
754:L10a140
714:(11n42)
708:(11n34)
676:Endless
301:1472978
219:Menasco
110:reduced
1199:Writhe
1169:Ribbon
1004:HOMFLY
847:Unlink
807:Unknot
782:Square
777:Granny
513:
436:
414:
404:
347:
325:(11).
299:
289:
184:flypes
160:writhe
1189:Twist
1174:Slice
1129:Berge
1117:Other
1088:Flype
1026:Prime
1009:Jones
969:Genus
799:Torus
613:links
609:knots
511:S2CID
493:arXiv
455:(PDF)
345:S2CID
327:arXiv
306:p. 32
169:and D
1194:Wild
1159:Knot
1063:and
1050:and
1031:list
862:Hopf
611:and
434:ISBN
402:ISBN
287:ISBN
229:for
206:and
198:and
38:link
34:knot
32:, a
1179:Sum
700:161
698:(10
503:doi
467:doi
412:Zbl
337:doi
323:166
279:doi
225:'s
75:).
71:, 8
67:, 8
36:or
28:In
1253::
879:(4
864:(2
849:(0
839:(7
829:(5
819:(3
809:(0
741:(6
726:(5
690:18
688:(8
678:(7
652:(6
642:(5
632:(4
556:.
537:.
509:.
501:.
489:88
487:.
463:23
461:.
457:.
410:.
369:.
343:.
335:.
321:.
297:MR
295:,
285:,
245:.
194:,
101:.
73:21
69:20
65:19
888:)
884:1
873:)
869:1
858:)
854:1
843:)
841:1
833:)
831:1
823:)
821:1
813:)
811:1
750:)
746:2
735:)
731:1
702:)
692:)
682:)
680:4
670:3
668:6
662:2
660:6
656:)
654:1
646:)
644:2
636:)
634:1
615:)
607:(
597:e
590:t
583:v
562:.
543:.
517:.
505::
495::
475:.
469::
442:.
418:.
375:.
351:.
339::
329::
281::
179:2
175:1
171:2
167:1
162:.
24:8
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