279:
270:
261:
208:
138:
1135:
1147:
31:
217:
is not tricolorable. In the diagram shown, it has four strands with each pair of strands meeting at some crossing. If three of the strands had the same color, then all strands would be forced to be the same color. Otherwise each of these four strands must have a distinct color. Since tricolorability
294:
Because tricolorability is a binary classification (a link is either tricolorable or not*), it is a relatively weak invariant. The composition of a tricolorable knot with another knot is always tricolorable. A way to strengthen the invariant is to count the number of possible 3-colorings. In this
295:
case, the rule that at least two colors are used is relaxed and now every link has at least three 3-colorings (just color every arc the same color). In this case, a link is 3-colorable if it has more than three 3-colorings.
101:
Some references state instead that all three colors must be used. For a knot, this is equivalent to the definition above; however, for a link it is not.
1178:
512:
443:
1080:
999:
242:. Since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant of tame knots.
368:
147:
is tricolorable. In this coloring the three strands at every crossing have three different colors. Coloring one but not both of the
546:
344:
214:
109:
989:
994:
865:
128:
a knot in accordance of the rules of tricolorability. By convention, knot theorists use the colors red, green, and blue.
566:
628:
634:
698:
693:
505:
144:
1183:
278:
310:/link denoted by (m,n) is tricolorable, then so are (j*m,i*n) and (i*n,j*m) for any natural numbers i and j.
1173:
826:
269:
1151:
260:
1040:
1009:
58:
is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an
870:
116:
on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is."
90:
if each strand of the knot diagram can be colored one of three colors, subject to the following rules:
97:
2. At each crossing, the three incident strands are either all the same color or all different colors.
1139:
910:
498:
947:
930:
207:
968:
915:
529:
525:
231:
113:
55:
1065:
1014:
964:
920:
880:
875:
793:
472:
449:
439:
374:
364:
239:
1100:
925:
821:
556:
151:
all red would also give an admissible coloring. The true lover's knot is also tricolorable.
1060:
1024:
905:
860:
853:
743:
655:
538:
411:
235:
63:
475:
415:
1120:
1019:
981:
900:
813:
688:
680:
640:
324:
319:
227:
105:
59:
86:
will be a piece of the string that goes from one undercrossing to the next. A knot is
1167:
1055:
843:
836:
831:
137:
1070:
1050:
954:
937:
733:
670:
148:
83:
35:
753:
592:
584:
576:
377:
298:
Any separable link with a tricolorable separable component is also tricolorable.
17:
1085:
848:
622:
602:
521:
490:
47:
43:
1105:
1090:
1045:
942:
895:
890:
885:
715:
612:
307:
435:
The knot book: an elementary introduction to the mathematical theory of knots
433:
1110:
778:
480:
453:
382:
244:
218:
is a knot invariant, none of its other diagrams can be tricolored either.
1095:
705:
30:
104:"The trefoil knot and trivial 2-link are tricolorable, but the unknot,
70:
is not tricolorable, any tricolorable knot is necessarily nontrivial.
1115:
763:
723:
67:
1004:
438:. Providence, R.I: American Mathematical Society. pp. 22–27.
125:
62:, and hence can be used to distinguish between two different (non-
29:
1075:
494:
277:
268:
259:
112:
are not. If the projection of a knot is tricolorable, then
154:
Tricolorable knots with less than nine crossings include 6
1033:
977:
812:
714:
679:
537:
238:. This can be proven for tame knots by examining
506:
8:
355:
353:
513:
499:
491:
234:that remains constant regardless of any
94:1. At least two colors must be used, and
361:CRC Concise Encyclopedia of Mathematics
336:
254:Reidemeister Move III is tricolorable.
251:Reidemeister Move II is tricolorable.
7:
1146:
398:Gilbert, N.D. and Porter, T. (1994)
248:Reidemeister Move I is tricolorable.
345:Knot Theory Week 2: Tricolorability
230:, which is a property of a knot or
66:) knots. In particular, since the
202:Example of a non-tricolorable knot
25:
1145:
1134:
1133:
416:Knots: a handout for mathcircles
206:
136:
1000:Dowker–Thistlethwaite notation
347:(January 20, 2015), Section 3.
132:Example of a tricolorable knot
1:
124:Here is an example of how to
1179:Tricolorable knots and links
359:Weisstein, Eric W. (2010).
1200:
363:, Second Edition, p.3045.
1129:
990:Alexander–Briggs notation
74:Rules of tricolorability
1081:List of knots and links
629:Kinoshita–Terasaka knot
27:Property in knot theory
487:Accessed: May 5, 2013.
476:"Three-Colorable Knot"
389:Accessed: May 5, 2013.
282:
273:
264:
226:Tricolorability is an
39:
871:Finite type invariant
432:Adams, Colin (2004).
281:
272:
263:
33:
1041:Alexander's theorem
343:Xaoyu Qiao, E. L.,
473:Weisstein, Eric W.
414:(February 2003). "
400:Knots and Surfaces
375:Weisstein, Eric W.
283:
274:
265:
240:Reidemeister moves
114:Reidemeister moves
40:
1161:
1160:
1015:Reidemeister move
881:Khovanov homology
876:Hyperbolic volume
445:978-0-8218-3678-1
287:
286:
228:isotopy invariant
222:Isotopy invariant
215:figure-eight knot
110:figure-eight knot
78:In these rules a
60:isotopy invariant
18:Tricolorable knot
16:(Redirected from
1191:
1149:
1148:
1137:
1136:
1101:Tait conjectures
804:
803:
789:
788:
774:
773:
666:
665:
651:
650:
635:(−2,3,7) pretzel
515:
508:
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492:
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458:
457:
429:
423:
412:Bestvina, Mladen
409:
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210:
140:
21:
1199:
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1190:
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1184:Knot invariants
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1163:
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1157:
1125:
1029:
995:Conway notation
979:
973:
960:Tricolorability
808:
802:
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759:
749:
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729:
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689:Composite knots
675:
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661:
660:
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656:Borromean rings
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618:
608:
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471:
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465:Further reading
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236:ambient isotopy
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177:
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161:
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134:
122:
76:
52:tricolorability
28:
23:
22:
15:
12:
11:
5:
1197:
1195:
1187:
1186:
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1174:Graph coloring
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1165:
1159:
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1127:
1126:
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1123:
1121:Surgery theory
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1113:
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1103:
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1063:
1058:
1053:
1048:
1043:
1037:
1035:
1031:
1030:
1028:
1027:
1022:
1020:Skein relation
1017:
1012:
1007:
1002:
997:
992:
986:
984:
975:
974:
972:
971:
965:Unknotting no.
962:
957:
952:
951:
950:
940:
935:
934:
933:
928:
923:
918:
913:
903:
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517:
510:
503:
495:
489:
488:
466:
463:
460:
459:
444:
424:
404:
391:
378:"Tricolorable"
349:
335:
334:
332:
329:
328:
327:
325:Graph coloring
322:
320:Fox n-coloring
315:
312:
303:
302:In torus knots
300:
291:
288:
285:
284:
275:
266:
256:
255:
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249:
223:
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200:
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191:
187:
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133:
130:
121:
118:
106:Whitehead link
99:
98:
95:
75:
72:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1196:
1185:
1182:
1180:
1177:
1175:
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1169:
1154:
1153:
1144:
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1128:
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1082:
1079:
1077:
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1067:
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1059:
1057:
1056:Conway sphere
1054:
1052:
1049:
1047:
1044:
1042:
1039:
1038:
1036:
1032:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
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1003:
1001:
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996:
993:
991:
988:
987:
985:
983:
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966:
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961:
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956:
953:
949:
946:
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939:
936:
932:
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927:
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922:
919:
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914:
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904:
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869:
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855:
852:
851:
850:
847:
845:
842:
838:
835:
834:
833:
830:
828:
827:Arf invariant
825:
823:
820:
819:
817:
815:
811:
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421:
420:Math.Utah.edu
417:
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370:
369:9781420035223
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241:
237:
233:
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209:
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199:
152:
150:
149:trefoil knots
146:
141:
139:
131:
129:
127:
119:
117:
115:
111:
107:
102:
96:
93:
92:
91:
89:
85:
81:
73:
71:
69:
65:
61:
57:
53:
49:
45:
37:
34:A tricolored
32:
19:
1150:
1138:
1066:Double torus
1051:Braid theory
959:
866:Crossing no.
861:Crosscap no.
547:Figure-eight
479:
434:
427:
419:
407:
399:
394:
381:
371:. quoted at
360:
339:
305:
297:
293:
225:
212:
205:
153:
142:
135:
123:
103:
100:
88:tricolorable
87:
84:knot diagram
79:
77:
51:
44:mathematical
41:
36:trefoil knot
901:Linking no.
822:Alternating
623:Conway knot
603:Carrick mat
557:Three-twist
522:Knot theory
145:granny knot
48:knot theory
1168:Categories
1061:Complement
1025:Tabulation
982:operations
906:Polynomial
896:Link group
891:Knot group
854:Invertible
832:Bridge no.
814:Invariants
744:Cinquefoil
613:Perko pair
539:Hyperbolic
308:torus knot
290:Properties
955:Stick no.
911:Alexander
849:Chirality
794:Solomon's
754:Septafoil
681:Satellite
641:Whitehead
567:Stevedore
481:MathWorld
383:MathWorld
46:field of
1140:Category
1010:Mutation
978:Notation
931:Kauffman
844:Brunnian
837:2-bridge
706:Knot sum
637:(12n242)
454:55633800
314:See also
120:Examples
64:isotopic
1152:Commons
1071:Fibered
969:problem
938:Pretzel
916:Bracket
734:Trefoil
671:L10a140
631:(11n42)
625:(11n34)
593:Endless
331:Sources
306:If the
194:, and 8
42:In the
1116:Writhe
1086:Ribbon
921:HOMFLY
764:Unlink
724:Unknot
699:Square
694:Granny
452:
442:
402:, p. 8
367:
108:, and
80:strand
68:unknot
50:, the
1106:Twist
1091:Slice
1046:Berge
1034:Other
1005:Flype
943:Prime
926:Jones
886:Genus
716:Torus
530:links
526:knots
126:color
82:in a
54:of a
1111:Wild
1076:Knot
980:and
967:and
948:list
779:Hopf
528:and
450:OCLC
440:ISBN
365:ISBN
232:link
213:The
143:The
56:knot
1096:Sum
617:161
615:(10
418:",
198:.
190:, 8
186:, 8
182:, 8
178:, 8
174:, 8
170:, 8
166:, 8
162:, 7
158:, 7
1170::
796:(4
781:(2
766:(0
756:(7
746:(5
736:(3
726:(0
658:(6
643:(5
607:18
605:(8
595:(7
569:(6
559:(5
549:(4
478:.
448:.
380:.
352:^
196:21
192:20
188:19
184:18
180:15
176:11
172:10
805:)
801:1
790:)
786:1
775:)
771:1
760:)
758:1
750:)
748:1
740:)
738:1
730:)
728:1
667:)
663:2
652:)
648:1
619:)
609:)
599:)
597:4
587:3
585:6
579:2
577:6
573:)
571:1
563:)
561:2
553:)
551:1
532:)
524:(
514:e
507:t
500:v
484:.
456:.
422:.
386:.
168:5
164:7
160:4
156:1
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
