List of prime knots

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In the mathematical theory of knots, prime knots are those knots that are indecomposable under the operation of knot sum. They are listed here for quick comparison of their properties and varied naming schemes.

Table of prime knots[edit]

Six or fewer crossings[edit]

Name Picture Alexander- Briggs-Rolfsen Dowker-

Thistlethwaite

Dowker notation Conway notation
Unknot Blue Unknot.png 01 0a1
Trefoil knot Blue Trefoil Knot.png 31 3a1 4 6 2 [3]
Figure-eight knot Blue Figure-Eight Knot.png 41 4a1 4 6 8 2 [22]
Cinquefoil knot Blue Cinquefoil Knot.png 51 5a2 6 8 10 2 4 [5]
Three-twist knot Blue Three-Twist Knot.png 52 5a1 4 8 10 2 6 [32]
Stevedore knot Blue Stevedore Knot.png 61 6a3 4 8 12 10 2 6 [42]
62 knot Blue 6 2 Knot.png 62 6a2 4 8 10 12 2 6 [312]
63 knot Blue 6 3 Knot.png 63 6a1 4 8 10 2 12 6 [2112]

Seven crossings[edit]

Picture Alexander-

Briggs-Rolfsen

Dowker-

Thistlethwaite

Dowker notation Conway notation
Blue 7 1 Knot.png 71 knot 7a7 8 10 12 14 2 4 6 [7]
Blue 7 2 Knot.png 72 7a4 4 10 14 12 2 8 6 [52]
KnotGraph7 3.svg 73 7a5 6 10 12 14 2 4 8 [43]
Celtic-knot-linear-7crossings.svg 74 knot 7a6 6 10 12 14 4 2 8 [313]
KnotGraph7 5.svg 75 7a3 4 10 12 14 2 8 6 [322]
KnotGraph7 6.svg 76 7a2 4 8 12 2 14 6 10 [2212]
KnotGraph7 7.svg 77 7a1 4 8 10 12 2 14 6 [21112]

Eight crossings[edit]

Picture Alexander-

Briggs-Rolfsen

Dowker-

Thistlethwaite

Dowker notation Conway notation
Blue 8 1 Knot.png 81 8a­11 4 10 16 14 12 2 8 6 [62]
82 8a8 4 10 12 14 16 2 6 8 [512]
83 8a­18 6 12 10 16 14 4 2 8 [44]
84 8a­17 6 10 12 16 14 4 2 8 [413]
85 8a­13 6 8 12 2 14 16 4 10 [3,3,2]
86 8a­10 4 10 14 16 12 2 8 6 [332]
87 8a6 4 10 12 14 2 16 6 8 [4112]
88 8a4 4 8 12 2 16 14 6 10 [2312]
89 8a­16 6 10 12 14 16 4 2 8 [3113]
810 8a3 4 8 12 2 14 16 6 10 [3,21,2]
811 8a9 4 10 12 14 16 2 8 6 [3212]
8crossings-rose-limacon-knot.svg 812 8a5 4 8 14 10 2 16 6 12 [2222]
813 8a7 4 10 12 14 2 16 8 6 [31112]
814 8a1 4 8 10 14 2 16 6 12 [22112]
8crossings-two-trefoils.svg 815 8a2 4 8 12 2 14 6 16 10 [21,21,2]
8-16 knot theory square.svg 816 8a­15 6 8 14 12 4 16 2 10 [.2.20]
8 17 Knot.svg 817 8a­14 6 8 12 14 4 16 2 10 [.2.2]
8crossing-symmetrical.svg 818 8a­12 6 8 10 12 14 16 2 4 [8*]
8crossing-symmetrical-nonalternating.svg 819 8n3 4 8 -12 2 -14 -16 -6 -10 [3,3,2-]
Knot 8 20.svg 820 8n1 4 8 -12 2 -14 -6 -16 -10 [3,21,2-]
Lissajous 8 21 Knot.png 821 8n2 4 8 -12 2 14 -6 16 10 [21,21,2-]

Nine crossings[edit]

Picture Alexander-

Briggs-Rolfsen

Dowker-

Thistlethwaite

Dowker notation Conway notation
9-2 star polygon interlaced.svg 91 9a­41 10 12 14 16 18 2 4 6 8 [9]
92 knot 9a­27 4 12 18 16 14 2 10 8 6 [72]
93 9a­38 8 12 14 16 18 2 4 6 10 [63]
94 9a­35 6 12 14 18 16 2 4 10 8 [54]
95 9a­36 6 12 14 18 16 4 2 10 8 [513]
96 9a­23 4 12 14 16 18 2 10 6 8 [522]
97 9a­26 4 12 16 18 14 2 10 8 6 [342]
98 9a8 4 8 14 2 18 16 6 12 10 [2412]
99 9a­33 6 12 14 16 18 2 4 10 8 [423]
910 9a­39 8 12 14 16 18 2 6 4 10 [333]
911 9a­20 4 10 14 16 12 2 18 6 8 [4122]
912 9a­22 4 10 16 14 2 18 8 6 12 [4212]
913 9a­34 6 12 14 16 18 4 2 10 8 [3213]
914 9a­17 4 10 12 16 14 2 18 8 6 [41112]
915 9a­10 4 8 14 10 2 18 16 6 12 [2322]
916 9a­25 4 12 16 18 14 2 8 10 6 [3,3,2+]
917 9a­14 4 10 12 14 16 2 6 18 8 [21312]
918 9a­24 4 12 14 16 18 2 10 8 6 [3222]
919 9a3 4 8 10 14 2 18 16 6 12 [23112]
920 9a­19 4 10 14 16 2 18 8 6 12 [31212]
921 9a­21 4 10 14 16 12 2 18 8 6 [31122]
922 9a2 4 8 10 14 2 16 18 6 12 [211,3,2]
9crossing-knot symmetrical grid.svg 923 9a­16 4 10 12 16 2 8 18 6 14 [22122]
924 9a7 4 8 14 2 16 18 6 12 10 [3,21,2+]
925 9a4 4 8 12 2 16 6 18 10 14 [22,21,2]
926 9a­15 4 10 12 14 16 2 18 8 6 [311112]
927 9a­12 4 10 12 14 2 18 16 6 8 [212112]
928 9a5 4 8 12 2 16 14 6 18 10 [21,21,2+]
929 9a­31 6 10 14 18 4 16 8 2 12 [.2.20.2]
930 9a1 4 8 10 14 2 16 6 18 12 [211,21,2]
931 9a­13 4 10 12 14 2 18 16 8 6 [2111112]
932 9a6 4 8 12 14 2 16 18 10 6 [.21.20]
933 9a­11 4 8 14 12 2 16 18 10 6 [.21.2]
934 9a­28 6 8 10 16 14 18 4 2 12 [8*20]
9crossings-threesymmetric-other.svg 935 9a­40 8 12 16 14 18 4 2 6 10 [3,3,3]
936 9a9 4 8 14 10 2 16 18 6 12 [22,3,2]
937 9a­18 4 10 14 12 16 2 6 18 8 [3,21,21]
938 9a­30 6 10 14 18 4 16 2 8 12 [.2.2.2]
939 9a­32 6 10 14 18 16 2 8 4 12 [2:2:20]
Knot-9crossings-symmetrical.svg 940 9a­27 6 16 14 12 4 2 18 10 8 [9*]
9crossings-decorative-knot-threefold-incircle.svg 941 9a­29 6 10 14 12 16 2 18 4 8 [20:20:20]
942 9n4 4 8 10 −14 2 −16 −18 −6 −12 [22,3,2−]
943 9n3 4 8 10 14 2 −16 6 −18 −12 [211,3,2−]
944 9n1 4 8 10 −14 2 −16 −6 −18 −12 [22,21,2−]
945 9n2 4 8 10 −14 2 16 −6 18 12 [211,21,2−]
946 9n5 4 10 −14 −12 −16 2 −6 −18 −8 [3,3,21−]
9-crossing non-alternating 3-symmetrical.svg 947 9n7 6 8 10 16 14 −18 4 2 −12 [8*-20]
948 9n6 4 10 −14 −12 16 2 −6 18 8 [21,21,21−]
949 9n8 6 -10 −14 12 −16 −2 18 −4 −8 [−20:−20:−20]

Ten crossings[edit]

Picture Alexander-

Briggs-Rolfsen

Dowker-

Thistlethwaite

Dowker notation Conway notation
101 knot 10a­75 4 12 20 18 16 14 2 10 8 6 [82]
102 10a­59 4 12 14 16 18 20 2 6 8 10 [712]
103 10a­­117 6 14 12 20 18 16 4 2 10 8 [64]
104 10a­­113 6 12 14 20 18 16 4 2 10 8 [613]
105 10a­56 4 12 14 16 18 2 20 6 8 10 [6112]
106 10a­70 4 12 16 18 20 14 2 10 6 8 [532]
107 10a­65 4 12 14 18 16 20 2 10 8 6 [5212]
108 10a­­114 6 14 12 16 18 20 4 2 8 10 [514]
109 10a­­110 6 12 14 16 18 20 4 2 8 10 [5113]
1010 10a­64 4 12 14 18 16 2 20 10 8 6 [51112]
1011 10a­­116 6 14 12 18 20 16 4 2 10 8 [433]
1012 10a­43 4 10 14 16 2 20 18 6 8 12 [4312]
1013 10a­54 4 10 18 16 12 2 20 8 6 14 [4222]
1014 10a­33 4 10 12 16 18 2 20 6 8 14 [42112]
1015 10a­68 4 12 16 18 14 2 10 20 6 8 [4132]
1016 10a­­115 6 14 12 16 18 20 4 2 10 8 [4123]
1017 10a­­107 6 12 14 16 18 2 4 20 8 10 [4114]
1018 10a­63 4 12 14 18 16 2 10 20 8 6 [41122]
1019 10a­­108 6 12 14 16 18 2 4 20 10 8 [41113]
1020 10a­74 4 12 18 20 16 14 2 10 8 6 [352]
1021 10a­60 4 12 14 16 18 20 2 6 10 8 [3412]
1022 10a­­112 6 12 14 18 20 16 4 2 10 8 [3313]
1023 10a­57 4 12 14 16 18 2 20 6 10 8 [33112]
1024 10a­71 4 12 16 18 20 14 2 10 8 6 [3232]
Knot-10-25-sm.png 1025 10a­61 4 12 14 16 18 20 2 10 8 6 [32212]
1026 10a­­111 6 12 14 16 18 20 4 2 10 8 [32113]
1027 10a­58 4 12 14 16 18 2 20 10 8 6 [321112]
1028 10a­44 4 10 14 16 2 20 18 8 6 12 [31312]
1029 10a­53 4 10 16 18 12 2 20 8 6 14 [31222]
1030 10a­34 4 10 12 16 18 2 20 8 6 14 [312112]
1031 10a­69 4 12 16 18 14 2 10 20 8 6 [31132]
1032 10a­55 4 12 14 16 18 2 10 20 8 6 [311122]
1033 10a­­109 6 12 14 16 18 4 2 20 10 8 [311113]
1034 10a­19 4 8 14 2 20 18 16 6 12 10 [2512]
1035 10a­23 4 8 16 10 2 20 18 6 14 12 [2422]
1036 10a5 4 8 10 16 2 20 18 6 14 12 [24112]
1037 10a­49 4 10 16 12 2 8 20 18 6 14 [2332]
1038 10a­29 4 10 12 16 2 8 20 18 6 14 [23122]
1039 10a­26 4 10 12 14 18 2 6 20 8 16 [22312]
1040 10a­30 4 10 12 16 2 20 6 18 8 14 [222112]
1041 10a­35 4 10 12 16 20 2 8 18 6 14 [221212]
1042 10a­31 4 10 12 16 2 20 8 18 6 14 [2211112]
1043 10a­52 4 10 16 14 2 20 8 18 6 12 [212212]
1044 10a­32 4 10 12 16 14 2 20 18 8 6 [2121112]
1045 10a­25 4 10 12 14 16 2 20 18 8 6 [21111112]
1046 10a­81 6 8 14 2 16 18 20 4 10 12 [5,3,2]
1047 10a­15 4 8 14 2 16 18 20 6 10 12 [5,21,2]
1048 10a­79 6 8 14 2 16 18 4 20 10 12 [41,3,2]
1049 10a­13 4 8 14 2 16 18 6 20 10 12 [41,21,2]
1050 10a­82 6 8 14 2 16 18 20 4 12 10 [32,3,2]
1051 10a­16 4 8 14 2 16 18 20 6 12 10 [32,21,2]
1052 10a­80 6 8 14 2 16 18 4 20 12 10 [311,3,2]
1053 10a­14 4 8 14 2 16 18 6 20 12 10 [311,21,2]
1054 10a­48 4 10 16 12 2 8 18 20 6 14 [23,3,2]
1055 10a9 4 8 12 2 16 6 20 18 10 14 [23,21,2]
1056 10a­28 4 10 12 16 2 8 18 20 6 14 [221,3,2]
1057 10a6 4 8 12 2 14 18 6 20 10 16 [221,21,2]
1058 10a­20 4 8 14 10 2 18 6 20 12 16 [22,22,2]
10-59 knot theory square.svg 1059 10a2 4 8 10 14 2 18 6 20 12 16 [22,211,2]
10-60 knot theory square.svg 1060 10a1 4 8 10 14 2 16 18 6 20 12 [211,211,2]
1061 10a­­123 8 10 16 14 2 18 20 6 4 12 [4,3,3]
1062 10a­41 4 10 14 16 2 18 20 6 8 12 [4,3,21]
1063 10a­51 4 10 16 14 2 18 8 6 20 12 [4,21,21]
1064 10a­­122 8 10 14 16 2 18 20 6 4 12 [31,3,3]
1065 10a­42 4 10 14 16 2 18 20 8 6 12 [31,3,21]
1066 10a­40 4 10 14 16 2 18 8 6 20 12 [31,21,21]
1067 10a­37 4 10 14 12 18 2 6 20 8 16 [22,3,21]
1068 10a­67 4 12 16 14 18 2 20 6 10 8 [211,3,3]
1069 10a­38 4 10 14 12 18 2 16 6 20 8 [211,21,21]
1070 10a­22 4 8 16 10 2 18 20 6 14 12 [22,3,2+]
1071 10a­10 4 8 12 2 18 14 6 20 10 16 [22,21,2+]
1072 10a4 4 8 10 16 2 18 20 6 14 12 [211,3,2+]
1073 10a3 4 8 10 14 2 18 16 6 20 12 [211,21,2+]
1074 10a­62 4 12 14 16 20 18 2 8 6 10 [3,3,21+]
Vodicka knot modified.svg 1075 10a­27 4 10 12 14 18 2 16 6 20 8 [21,21,21+]
1076 10a­73 4 12 18 20 14 16 2 10 8 6 [3,3,2++]
1077 10a­18 4 8 14 2 18 20 16 6 12 10 [3,21,2++]
1078 10a­17 4 8 14 2 18 16 6 12 20 10 [21,21,2++]
1079 10a­78 6 8 12 2 16 4 18 20 10 14 [(3,2)(3,2)]
1080 10a8 4 8 12 2 16 6 18 20 10 14 [(3,2)(21,2)]
1081 10a7 4 8 12 2 16 6 18 10 20 14 [(21,2)(21,2)]
1082 10a­83 6 8 14 16 4 18 20 2 10 12 [.4.2]
1083 10a­84 6 8 16 14 4 18 20 2 12 10 [.31.20]
1084 10a­50 4 10 16 14 2 8 18 20 12 6 [.22.2]
1085 10a­86 6 8 16 14 4 18 20 2 10 12 [.4.20]
1086 10a­87 6 8 14 16 4 18 20 2 12 10 [.31.2]
1087 10a­39 4 10 14 16 2 8 18 20 12 6 [.22.20]
1088 10a­11 4 8 12 14 2 16 20 18 10 6 [.21.21]
1089 10a­21 4 8 14 12 2 16 20 18 10 6 [.21.210]
1090 10a­92 6 10 14 2 16 20 18 8 4 12 [.3.2.2]
1091 10a­­106 6 10 20 14 16 18 4 8 2 12 [.3.2.20]
1092 10a­46 4 10 14 18 2 16 8 20 12 6 [.21.2.20]
1093 10a­­101 6 10 16 20 14 4 18 2 12 8 [.3.20.2]
1094 10a­91 6 10 14 2 16 18 20 8 4 12 [.30.2.2]
1095 10a­47 4 10 14 18 2 16 20 8 12 6 [.210.2.2]
1096 10a­24 4 8 18 12 2 16 20 6 10 14 [.2.21.2]
1097 10a­12 4 8 12 18 2 16 20 6 10 14 [.2.210.2]
1098 10a­96 6 10 14 18 2 16 20 4 8 12 [.2.2.2.20]
1099 10a­­103 6 10 18 14 2 16 20 8 4 12 [.2.2.20.20]
10100 10a­­104 6 10 18 14 16 4 20 8 2 12 [3:2:2]
10101 10a­45 4 10 14 18 2 16 6 20 8 12 [21:2:2]
10102 10a­97 6 10 14 18 16 4 20 2 8 12 [3:2:20]
10103 10a­­105 6 10 18 16 14 4 20 8 2 12 [30:2:2]
10104 10a­­118 6 16 12 14 18 4 20 2 8 10 [3:20:20]
10105 10a­72 4 12 16 20 18 2 8 6 10 14 [21:20:20]
10106 10a­95 6 10 14 16 18 4 20 2 8 12 [30:2:20]
10107 10a­66 4 12 16 14 18 2 8 20 10 6 [210:2:20]
10108 10a­­119 6 16 12 14 18 4 20 2 10 8 [30:20:20]
10109 10a­93 6 10 14 16 2 18 4 20 8 12 [2.2.2.2]
10110 10a­­100 6 10 16 20 14 2 18 4 8 12 [2.2.2.20]
10111 10a­98 6 10 16 14 2 18 8 20 4 12 [2.2.20.2]
10112 10a­76 6 8 10 14 16 18 20 2 4 12 [8*3]
10113 10a­36 4 10 14 12 2 16 18 20 8 6 [8*21]
10114 10a­77 6 8 10 14 16 20 18 2 4 12 [8*30]
10115 10a­94 6 10 14 16 4 18 2 20 12 8 [8*20.20]
Triquetra-heart-knot.svg 10116 10a­­120 6 16 18 14 2 4 20 8 10 12 [8*2:2]
10117 10a­99 6 10 16 14 18 4 20 2 12 8 [8*2:20]
10118 10a­88 6 8 18 14 16 4 20 2 10 12 [8*2:.2]
10119 10a­85 6 8 14 18 16 4 20 10 2 12 [8*2:.20]
Two-trefoils-on-loop doubly-interlinked 10crossings.svg 10120 10a­­102 6 10 18 12 4 16 20 8 2 14 [8*20::20]
10121 10a­90 6 10 12 20 18 16 8 2 4 14 [9*20]
10crossings-two-triquetras-joined.svg 10122 10a­89 6 10 12 14 18 16 20 2 4 8 [9*.20]
Floral fivefold knot green (geometry).svg 10123 10a­­121 8 10 12 14 16 18 20 2 4 6 [10*]
10124 10n­21 4 8 -14 2 -16 -18 -20 -6 -10 -12 [5,3,2-]
10125 10n­15 4 8 14 2 -16 -18 6 -20 -10 -12 [5,21,2-]
10126 10n­17 4 8 -14 2 -16 -18 -6 -20 -10 -12 [41,3,2-]
10127 10n­16 4 8 -14 2 16 18 -6 20 10 12 [41,21,2-]
10128 10n­22 4 8 -14 2 -16 -18 -20 -6 -12 -10 [32,3,2-]
10129 10n­18 4 8 14 2 -16 -18 6 -20 -12 -10 [32,21,-2]
10130 10n­20 4 8 -14 2 -16 -18 -6 -20 -12 -10 [311,3,2-]
10131 10n­19 4 8 -14 2 16 18 -6 20 12 10 [311,21,2-]
Knot-10-132-sm.png 10132 10n­13 4 8 -12 2 -16 -6 -20 -18 -10 -14 [23,3,2-]
10133 10n4 4 8 12 2 -14 -18 6 -20 -10 -16 [23,21,2-]
10134 10n6 4 8 -12 2 -14 -18 -6 -20 -10 -16 [221,3,2-]
10135 10n5 4 8 -12 2 14 18 -6 20 10 16 [221,21,2-]
10136 10n3 4 8 10 -14 2 -18 -6 -20 -12 -16 [22,22,2-]
10137 10n2 4 8 10 -14 2 -16 -18 -6 -20 -12 [22,211,2-]
10138 10n1 4 8 10 -14 2 16 18 -6 20 12 [211,211,2-]
10139 10n­27 4 10 -14 -16 2 -18 -20 -6 -8 -12 [4,3,3-]
10140 10n­29 4 10 -14 -16 2 18 20 -8 -6 12 [4,3,21-]
10141 10n­25 4 10 -14 -16 2 18 -8 -6 20 12 [4,21,21-]
10142 10n­30 4 10 -14 -16 2 -18 -20 -8 -6 -12 [31,3,3-]
10143 10n­26 4 10 -14 -16 2 -18 -8 -6 -20 -12 [31,3,21-]
10144 10n­28 4 10 14 16 2 -18 -20 8 6 -12 [31,21,21-]
10145 10n­14 4 8 -12 -18 2 -16 -20 -6 -10 -14 [22,3,3-]
10146 10n­23 4 8 -18 -12 2 -16 -20 -6 -10 -14 [22,21,21-]
10147 10n­24 4 10 -14 12 2 16 18 -20 8 -6 [211,3,21-]
10148 10n­12 4 8 -12 2 -16 -6 -18 -20 -10 -14 [(3,2)(3,2-)]
10149 10n­11 4 8 -12 2 16 -6 18 20 10 14 [(3,2)(21,2-)]
10150 10n9 4 8 -12 2 -16 -6 -18 -10 -20 -14 [(21,2)(3,2-)]
10151 10n8 4 8 -12 2 16 -6 18 10 20 14 [(21,2)(21,2-)]
10152 10n­36 6 8 12 2 -16 4 -18 -20 -10 -14 [(3,2)-(3,2)]
10153 10n­10 4 8 12 2 -16 6 -18 -20 -10 -14 [(3,2)-(21,2)]
10154 10n7 4 8 12 2 -16 6 -18 -10 -20 -14 [(21,2)-(21,2)]
10155 10n­39 6 10 14 16 18 4 -20 2 8 -12 [-3:2:2]
10156 10n­32 4 12 16 -14 18 2 -8 20 10 6 [-3:2:20]
10157 10n­42 6 -10 -18 14 -2 -16 20 8 -4 12 [-3:20:20]
10158 10n­41 6 -10 -16 14 -2 -18 8 20 -4 -12 [-30:2:2]
10159 10n­34 6 8 10 14 16 -18 -20 2 4 -12 [-30:2:20]
10160 10n­33 4 12 -16 -14 -18 2 -8 -20 -10 -6 [-30:20:20]
Ten onehundredandsixtyone.gif 10161[a] 10n­31 4 12 -16 14 -18 2 8 -20 -10 -6 [3:-20:-20]
10162[b] 10n­40 6 10 14 18 16 4 -20 2 8 -12 [-30:-20:-20]
10163[c] 10n­35 6 8 10 14 16 -20 -18 2 4 -12 [8*-30]
10164[d] 10n­38 6 -10 -12 14 -18 -16 20 -2 -4 -8 [8*2:-20]
10165[e] 10n­37 6 8 14 18 16 4 -20 10 2 -12 [8*2:.-20]

Notes[edit]

  1. ^ Originally listed as both 10161 and 10162 in the Rolfsen table. The error was discovered by Kenneth Perko (see Perko pair).
  2. ^ Listed as 10163 in the Rolfsen table.
  3. ^ Listed as 10164 in the Rolfsen table.
  4. ^ Listed as 10165 in the Rolfsen table.
  5. ^ Listed as 10166 in the Rolfsen table.

See also[edit]

External links[edit]