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20:
1673:
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112:
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149:
Six is the lowest stick number for any nontrivial knot. There are few knots whose stick number can be determined exactly. Gyo Taek Jin determined the stick number of a
217:
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518:
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138:
80:
60:
42:
that intuitively gives the smallest number of straight "sticks" stuck end to end needed to form a knot. Specifically, given any knot
1050:
812:; Brennan, Bevin M.; Greilsheimer, Deborah L.; Woo, Alexander K. (1997), "Stick numbers and composition of knots and links",
1618:
1537:
264:
792:
1084:
669:{\displaystyle {\frac {1}{2}}(7+{\sqrt {8\,{\text{c}}(K)+1}})\leq {\text{stick}}(K)\leq {\frac {3}{2}}(c(K)+1).}
85:
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1043:
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776:. An accessible introduction into the topic, also for readers with little mathematical background.
1364:
486:{\displaystyle {\text{stick}}(K_{1}\#K_{2})\leq {\text{stick}}(K_{1})+{\text{stick}}(K_{2})-3\,}
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1036:
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Jin, Gyo Taek (1997), "Polygon indices and superbridge indices of torus knots and links",
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8:
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1468:
363:
The same result was found independently around the same time by a research group around
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1004:
981:
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1028:
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Huh, Youngsik; Oh, Seungsang (2011), "An upper bound on stick number of knots",
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1374:
1369:
924:
Negami, Seiya (1991), "Ramsey theorems for knots, links and spatial graphs",
785:
The Knot Book: An elementary introduction to the mathematical theory of knots
840:
Calvo, Jorge
Alberto (2001), "Geometric knot spaces and polygonal isotopy",
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1316:
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16:
Smallest number of edges of an equivalent polygonal path for a knot
880:
New stick number bounds from random sampling of confined polygons
1653:
1301:
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1542:
682:, which has a crossing number of 3 and a stick number of 6.
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can be upper bounded by the stick numbers of the summands:
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926:Transactions of the American Mathematical Society
787:, Providence, RI: American Mathematical Society,
310:{\displaystyle \operatorname {stick} (T(p,q))=2q}
1703:
877:
769:"Why knot: knots, molecules and stick numbers"
1044:
878:Eddy, Thomas D.; Shonkwiler, Clayton (2019),
956:Journal of Knot Theory and its Ramifications
898:Journal of Knot Theory and its Ramifications
842:Journal of Knot Theory and its Ramifications
814:Journal of Knot Theory and its Ramifications
379:Square knot = trefoil + trefoil reflection.
1051:
1037:
678:These inequalities are both tight for the
967:
937:
887:
853:
583:
482:
367:, but for a smaller range of parameters.
107:{\displaystyle \operatorname {stick} (K)}
757:
374:
18:
114:, is the smallest number of edges of a
1704:
1023:KnotPlot Research and Development Site
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1032:
1019:Stick numbers for minimal stick knots
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354:{\displaystyle 2\leq p<q\leq 2p.}
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13:
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14:
1723:
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259:are not too far from each other:
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552:by the following inequalities:
144:
1538:Dowker–Thistlethwaite notation
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32:mathematical theory of knots
7:
500:The stick number of a knot
24:2,3 torus (or trefoil) knot
10:
1728:
26:has a stick number of six.
1667:
1571:
1528:Alexander–Briggs notation
1515:
1350:
1252:
1217:
1075:
978:10.1142/S0218216511008966
910:10.1142/S0218216597000170
864:10.1142/S0218216501000834
826:10.1142/S0218216597000121
370:
690:
1619:List of knots and links
1167:Kinoshita–Terasaka knot
219:in case the parameters
670:
546:
514:
487:
383:The stick number of a
380:
355:
311:
253:
233:
213:
212:{\displaystyle T(p,q)}
175:
134:
108:
76:
62:, the stick number of
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27:
1409:Finite type invariant
758:Introductory material
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234:
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176:
174:{\displaystyle (p,q)}
135:
109:
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57:
22:
556:
545:{\displaystyle c(K)}
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124:
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66:
46:
1579:Alexander's theorem
1001:Weisstein, Eric W.
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520:is related to its
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496:Related invariants
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1553:Reidemeister move
1419:Khovanov homology
1414:Hyperbolic volume
803:Research articles
749:Huh & Oh 2011
726:Adams et al. 1997
715:Adams et al. 1997
637:
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604:
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513:{\displaystyle K}
458:
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252:{\displaystyle q}
232:{\displaystyle p}
133:{\displaystyle K}
75:{\displaystyle K}
55:{\displaystyle K}
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1674:
1639:Tait conjectures
1342:
1341:
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1188:
1173:(−2,3,7) pretzel
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1712:Knot invariants
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1533:Conway notation
1517:
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1498:Tricolorability
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1194:Borromean rings
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810:Adams, Colin C.
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17:
12:
11:
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1503:Unknotting no.
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1005:"Stick number"
994:
993:External links
991:
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989:
962:(5): 741–747,
951:
932:(2): 527–541,
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904:(2): 281–289,
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116:polygonal path
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40:knot invariant
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1594:Conway sphere
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1366:
1365:Arf invariant
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1207:
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1180:
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1174:
1171:
1168:
1165:
1162:
1159:
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1142:
1139:
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1129:
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827:
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819:
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794:0-8218-3678-1
790:
786:
782:
778:
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773:Plus Magazine
770:
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127:
117:
98:
92:
89:
82:, denoted by
69:
49:
41:
37:
33:
25:
21:
1688:
1676:
1604:Double torus
1589:Braid theory
1492:
1404:Crossing no.
1399:Crosscap no.
1085:Figure-eight
1022:
1008:
959:
955:
929:
925:
901:
897:
879:
855:math/9904037
845:
841:
817:
813:
784:
781:Adams, C. C.
772:
767:(May 2001),
765:Adams, C. C.
736:
721:
710:
699:
680:trefoil knot
677:
499:
382:
362:
148:
145:Known values
36:stick number
35:
29:
1439:Linking no.
1360:Alternating
1161:Conway knot
1141:Carrick mat
1095:Three-twist
1060:Knot theory
741:Negami 1991
365:Colin Adams
118:equivalent
1599:Complement
1563:Tabulation
1520:operations
1444:Polynomial
1434:Link group
1429:Knot group
1392:Invertible
1370:Bridge no.
1352:Invariants
1282:Cinquefoil
1151:Perko pair
1077:Hyperbolic
969:1512.03592
889:1909.00917
745:Calvo 2001
686:References
183:torus knot
1493:Stick no.
1449:Alexander
1387:Chirality
1332:Solomon's
1292:Septafoil
1219:Satellite
1179:Whitehead
1105:Stevedore
1010:MathWorld
627:≤
610:≤
477:−
429:≤
413:#
340:≤
328:≤
272:
93:
1706:Category
1678:Category
1548:Mutation
1516:Notation
1469:Kauffman
1382:Brunnian
1375:2-bridge
1244:Knot sum
1175:(12n242)
783:(2004),
730:Jin 1997
704:Jin 1997
385:knot sum
1690:Commons
1609:Fibered
1507:problem
1476:Pretzel
1454:Bracket
1272:Trefoil
1209:L10a140
1169:(11n42)
1163:(11n34)
1131:Endless
986:2806342
948:1069741
918:1452441
872:1822491
834:1452436
30:In the
1654:Writhe
1624:Ribbon
1459:HOMFLY
1302:Unlink
1262:Unknot
1237:Square
1232:Granny
984:
946:
916:
870:
832:
791:
371:Bounds
34:, the
1644:Twist
1629:Slice
1584:Berge
1572:Other
1543:Flype
1481:Prime
1464:Jones
1424:Genus
1254:Torus
1068:links
1064:knots
964:arXiv
884:arXiv
850:arXiv
691:Notes
614:stick
457:stick
433:stick
396:stick
317:, if
269:stick
90:stick
38:is a
1649:Wild
1614:Knot
1518:and
1505:and
1486:list
1317:Hopf
1066:and
789:ISBN
334:<
239:and
1634:Sum
1155:161
1153:(10
1021:",
974:doi
934:doi
930:324
906:doi
860:doi
822:doi
120:to
1708::
1334:(4
1319:(2
1304:(0
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