List of prime knots explained

In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their properties and varied naming schemes.

Table of prime knots

Six or fewer crossings

NamePictureAlexander–
Briggs–
Rolfsen
Dowker–
Thistlethwaite
Dowker
notation
Conway
notation
crossinglist
Unknot010a1 - - 0
Trefoil knot313a14 6 2[3]123:123
Figure-eight knot414a14 6 8 2[22]1234:21431231\4324
Cinquefoil knot515a26 8 10 2 4[5]12345:12345
Three-twist knot525a14 8 10 2 6[32]12345:125431231\452354
Stevedore knot616a34 8 12 10 2 6[42]123456:2165431231\45632654
62 knot626a24 8 10 12 2 6[312]123456:2341651231\45632456
63 knot636a14 8 10 2 12 6[2112]123456:2361451231\45642356

1231\45236456

Seven crossings

PictureAlexander–
Briggs–
Rolfsen
Dowker–
Thistlethwaite
Dowker
notation
Conway
notation
crossinglist
717a78 10 12 14 2 4 6[7]1-7:1-7
727a44 10 14 12 2 8 6[52]1-7:127-3
737a56 10 12 14 2 4 8[43]
747a66 10 12 14 4 2 8[313]
757a34 10 12 14 2 8 6[322]
767a24 8 12 2 14 6 10[2212]
777a14 8 10 12 2 14 6[21112]

Eight crossings

PictureAlexander–
Briggs–
Rolfsen
Dowker–
Thistlethwaite
Dowker
notation
Conway
notation
818a­114 10 16 14 12 2 8 6[62]
828a84 10 12 14 16 2 6 8[512]
838a­186 12 10 16 14 4 2 8[44]
848a­176 10 12 16 14 4 2 8[413]
858a­136 8 12 2 14 16 4 10 [3,3,2]
868a­104 10 14 16 12 2 8 6[332]
878a64 10 12 14 2 16 6 8[4112]
888a44 8 12 2 16 14 6 10[2312]
898a­166 10 12 14 16 4 2 8[3113]
8108a34 8 12 2 14 16 6 10[3,21,2]
8118a94 10 12 14 16 2 8 6[3212]
8128a54 8 14 10 2 16 6 12[2222]
8138a74 10 12 14 2 16 8 6[31112]
8148a14 8 10 14 2 16 6 12[22112]
8158a24 8 12 2 14 6 16 10[21,21,2]
8168a­156 8 14 12 4 16 2 10[.2.20]
8178a­146 8 12 14 4 16 2 10[.2.2]
8188a­126 8 10 12 14 16 2 4[8*]
8198n34 8 -12 2 -14 -16 -6 -10[3,3,2-]
8208n14 8 -12 2 -14 -6 -16 -10[3,21,2-]
8218n24 8 -12 2 14 -6 16 10[21,21,2-]

Nine crossings

PictureAlexander–
Briggs–
Rolfsen
Dowker–
Thistlethwaite
Dowker
notation
Conway
notation
919a­4110 12 14 16 18 2 4 6 8[9]
929a­274 12 18 16 14 2 10 8 6[72]
939a­388 12 14 16 18 2 4 6 10[63]
949a­356 12 14 18 16 2 4 10 8[54]
959a­366 12 14 18 16 4 2 10 8[513]
969a­234 12 14 16 18 2 10 6 8[522]
979a­264 12 16 18 14 2 10 8 6[342]
989a84 8 14 2 18 16 6 12 10[2412]
999a­336 12 14 16 18 2 4 10 8[423]
9109a­398 12 14 16 18 2 6 4 10[333]
9119a­204 10 14 16 12 2 18 6 8[4122]
9129a­224 10 16 14 2 18 8 6 12[4212]
9139a­346 12 14 16 18 4 2 10 8[3213]
9149a­174 10 12 16 14 2 18 8 6[41112]
9159a­104 8 14 10 2 18 16 6 12[2322]
9169a­254 12 16 18 14 2 8 10 6[3,3,2+]
9179a­144 10 12 14 16 2 6 18 8[21312]
9189a­244 12 14 16 18 2 10 8 6[3222]
9199a34 8 10 14 2 18 16 6 12[23112]
9209a­194 10 14 16 2 18 8 6 12[31212]
9219a­214 10 14 16 12 2 18 8 6[31122]
9229a24 8 10 14 2 16 18 6 12[211,3,2]
9239a­164 10 12 16 2 8 18 6 14[22122]
9249a74 8 14 2 16 18 6 12 10[3,21,2+]
9259a44 8 12 2 16 6 18 10 14[22,21,2]
9269a­154 10 12 14 16 2 18 8 6[311112]
9279a­124 10 12 14 2 18 16 6 8[212112]
9289a54 8 12 2 16 14 6 18 10[21,21,2+]
9299a­316 10 14 18 4 16 8 2 12[.2.20.2]
9309a14 8 10 14 2 16 6 18 12[211,21,2]
9319a­134 10 12 14 2 18 16 8 6[2111112]
9329a64 8 12 14 2 16 18 10 6[.21.20]
9339a­114 8 14 12 2 16 18 10 6[.21.2]
9349a­286 8 10 16 14 18 4 2 12[8*20]
9359a­408 12 16 14 18 4 2 6 10[3,3,3]
9369a94 8 14 10 2 16 18 6 12[22,3,2]
9379a­184 10 14 12 16 2 6 18 8[3,21,21]
9389a­306 10 14 18 4 16 2 8 12[.2.2.2]
9399a­326 10 14 18 16 2 8 4 12[2:2:20]
9409a­276 16 14 12 4 2 18 10 8[9*]
9419a­296 10 14 12 16 2 18 4 8[20:20:20]
9429n44 8 10 -14 2 -16 -18 -6 -12[22,3,2−]
9439n34 8 10 14 2 -16 6 -18 -12[211,3,2−]
9449n14 8 10 -14 2 -16 -6 -18 -12[22,21,2−]
9459n24 8 10 -14 2 16 -6 18 12[211,21,2−]
9469n54 10 -14 -12 -16 2 -6 -18 -8[3,3,21−]
9479n76 8 10 16 14 -18 4 2 -12[8*-20]
9489n64 10 -14 -12 16 2 -6 18 8[21,21,21−]
9499n86 -10 -14 12 -16 -2 18 -4 -8[−20:−20:−20]

Ten crossings

PictureAlexander–
Briggs–
Rolfsen
Dowker–
Thistlethwaite
Dowker
notation
Conway
notation
10110a­754 12 20 18 16 14 2 10 8 6[82]
10210a­594 12 14 16 18 20 2 6 8 10[712]
10310a­­1176 14 12 20 18 16 4 2 10 8[64]
10410a­­1136 12 14 20 18 16 4 2 10 8[613]
10510a­564 12 14 16 18 2 20 6 8 10[6112]
10610a­704 12 16 18 20 14 2 10 6 8[532]
10710a­654 12 14 18 16 20 2 10 8 6[5212]
10810a­­1146 14 12 16 18 20 4 2 8 10[514]
10910a­­1106 12 14 16 18 20 4 2 8 10[5113]
101010a­644 12 14 18 16 2 20 10 8 6[51112]
101110a­­1166 14 12 18 20 16 4 2 10 8[433]
101210a­434 10 14 16 2 20 18 6 8 12[4312]
101310a­544 10 18 16 12 2 20 8 6 14[4222]
101410a­334 10 12 16 18 2 20 6 8 14[42112]
101510a­684 12 16 18 14 2 10 20 6 8[4132]
101610a­­1156 14 12 16 18 20 4 2 10 8[4123]
101710a­­1076 12 14 16 18 2 4 20 8 10[4114]
101810a­634 12 14 18 16 2 10 20 8 6[41122]
101910a­­1086 12 14 16 18 2 4 20 10 8[41113]
102010a­744 12 18 20 16 14 2 10 8 6[352]
102110a­604 12 14 16 18 20 2 6 10 8[3412]
102210a­­1126 12 14 18 20 16 4 2 10 8[3313]
102310a­574 12 14 16 18 2 20 6 10 8[33112]
102410a­714 12 16 18 20 14 2 10 8 6[3232]
102510a­614 12 14 16 18 20 2 10 8 6[32212]
102610a­­1116 12 14 16 18 20 4 2 10 8[32113]
102710a­584 12 14 16 18 2 20 10 8 6[321112]
102810a­444 10 14 16 2 20 18 8 6 12[31312]
102910a­534 10 16 18 12 2 20 8 6 14[31222]
103010a­344 10 12 16 18 2 20 8 6 14[312112]
103110a­694 12 16 18 14 2 10 20 8 6[31132]
103210a­554 12 14 16 18 2 10 20 8 6[311122]
103310a­­1096 12 14 16 18 4 2 20 10 8[311113]
103410a­194 8 14 2 20 18 16 6 12 10[2512]
103510a­234 8 16 10 2 20 18 6 14 12[2422]
103610a54 8 10 16 2 20 18 6 14 12[24112]
103710a­494 10 16 12 2 8 20 18 6 14[2332]
103810a­294 10 12 16 2 8 20 18 6 14[23122]
103910a­264 10 12 14 18 2 6 20 8 16[22312]
104010a­304 10 12 16 2 20 6 18 8 14[222112]
104110a­354 10 12 16 20 2 8 18 6 14[221212]
104210a­314 10 12 16 2 20 8 18 6 14[2211112]
104310a­524 10 16 14 2 20 8 18 6 12[212212]
104410a­324 10 12 16 14 2 20 18 8 6[2121112]
104510a­254 10 12 14 16 2 20 18 8 6[21111112]
104610a­816 8 14 2 16 18 20 4 10 12[5,3,2]
104710a­154 8 14 2 16 18 20 6 10 12[5,21,2]
104810a­796 8 14 2 16 18 4 20 10 12[41,3,2]
104910a­134 8 14 2 16 18 6 20 10 12[41,21,2]
105010a­826 8 14 2 16 18 20 4 12 10[32,3,2]
105110a­164 8 14 2 16 18 20 6 12 10[32,21,2]
105210a­806 8 14 2 16 18 4 20 12 10[311,3,2]
105310a­144 8 14 2 16 18 6 20 12 10[311,21,2]
105410a­484 10 16 12 2 8 18 20 6 14[23,3,2]
105510a94 8 12 2 16 6 20 18 10 14[23,21,2]
105610a­284 10 12 16 2 8 18 20 6 14[221,3,2]
105710a64 8 12 2 14 18 6 20 10 16[221,21,2]
105810a­204 8 14 10 2 18 6 20 12 16[22,22,2]
105910a24 8 10 14 2 18 6 20 12 16[22,211,2]
106010a14 8 10 14 2 16 18 6 20 12[211,211,2]
106110a­­1238 10 16 14 2 18 20 6 4 12[4,3,3]
106210a­414 10 14 16 2 18 20 6 8 12[4,3,21]
106310a­514 10 16 14 2 18 8 6 20 12[4,21,21]
106410a­­1228 10 14 16 2 18 20 6 4 12[31,3,3]
106510a­424 10 14 16 2 18 20 8 6 12[31,3,21]
106610a­404 10 14 16 2 18 8 6 20 12[31,21,21]
106710a­374 10 14 12 18 2 6 20 8 16[22,3,21]
106810a­674 12 16 14 18 2 20 6 10 8[211,3,3]
106910a­384 10 14 12 18 2 16 6 20 8[211,21,21]
107010a­224 8 16 10 2 18 20 6 14 12[22,3,2+]
107110a­104 8 12 2 18 14 6 20 10 16[22,21,2+]
107210a44 8 10 16 2 18 20 6 14 12[211,3,2+]
107310a34 8 10 14 2 18 16 6 20 12[211,21,2+]
107410a­624 12 14 16 20 18 2 8 6 10[3,3,21+]
107510a­274 10 12 14 18 2 16 6 20 8[21,21,21+]
107610a­734 12 18 20 14 16 2 10 8 6[3,3,2++]
107710a­184 8 14 2 18 20 16 6 12 10[3,21,2++]
107810a­174 8 14 2 18 16 6 12 20 10[21,21,2++]
107910a­786 8 12 2 16 4 18 20 10 14[(3,2)(3,2)]
108010a84 8 12 2 16 6 18 20 10 14[(3,2)(21,2)]
108110a74 8 12 2 16 6 18 10 20 14[(21,2)(21,2)]
108210a­836 8 14 16 4 18 20 2 10 12[.4.2]
108310a­846 8 16 14 4 18 20 2 12 10[.31.20]
108410a­504 10 16 14 2 8 18 20 12 6[.22.2]
108510a­866 8 16 14 4 18 20 2 10 12[.4.20]
108610a­876 8 14 16 4 18 20 2 12 10[.31.2]
108710a­394 10 14 16 2 8 18 20 12 6[.22.20]
108810a­114 8 12 14 2 16 20 18 10 6[.21.21]
108910a­214 8 14 12 2 16 20 18 10 6[.21.210]
109010a­926 10 14 2 16 20 18 8 4 12[.3.2.2]
109110a­­1066 10 20 14 16 18 4 8 2 12[.3.2.20]
109210a­464 10 14 18 2 16 8 20 12 6[.21.2.20]
109310a­­1016 10 16 20 14 4 18 2 12 8[.3.20.2]
109410a­916 10 14 2 16 18 20 8 4 12[.30.2.2]
109510a­474 10 14 18 2 16 20 8 12 6[.210.2.2]
109610a­244 8 18 12 2 16 20 6 10 14[.2.21.2]
109710a­124 8 12 18 2 16 20 6 10 14[.2.210.2]
109810a­966 10 14 18 2 16 20 4 8 12[.2.2.2.20]
109910a­­1036 10 18 14 2 16 20 8 4 12[.2.2.20.20]
1010010a­­1046 10 18 14 16 4 20 8 2 12[3:2:2]
1010110a­454 10 14 18 2 16 6 20 8 12[21:2:2]
1010210a­976 10 14 18 16 4 20 2 8 12[3:2:20]
1010310a­­1056 10 18 16 14 4 20 8 2 12[30:2:2]
1010410a­­1186 16 12 14 18 4 20 2 8 10[3:20:20]
1010510a­724 12 16 20 18 2 8 6 10 14[21:20:20]
1010610a­956 10 14 16 18 4 20 2 8 12[30:2:20]
1010710a­664 12 16 14 18 2 8 20 10 6[210:2:20]
1010810a­­1196 16 12 14 18 4 20 2 10 8[30:20:20]
1010910a­936 10 14 16 2 18 4 20 8 12[2.2.2.2]
1011010a­­1006 10 16 20 14 2 18 4 8 12[2.2.2.20]
1011110a­986 10 16 14 2 18 8 20 4 12[2.2.20.2]
1011210a­766 8 10 14 16 18 20 2 4 12[8*3]
1011310a­364 10 14 12 2 16 18 20 8 6[8*21]
1011410a­776 8 10 14 16 20 18 2 4 12[8*30]
1011510a­946 10 14 16 4 18 2 20 12 8[8*20.20]
1011610a­­1206 16 18 14 2 4 20 8 10 12[8*2:2]
1011710a­996 10 16 14 18 4 20 2 12 8[8*2:20]
1011810a­886 8 18 14 16 4 20 2 10 12[8*2:.2]
1011910a­856 8 14 18 16 4 20 10 2 12[8*2:.20]
1012010a­­1026 10 18 12 4 16 20 8 2 14[8*20::20]
1012110a­906 10 12 20 18 16 8 2 4 14[9*20]
1012210a­896 10 12 14 18 16 20 2 4 8[9*.20]
1012310a­­1218 10 12 14 16 18 20 2 4 6[10*]
1012410n­214 8 -14 2 -16 -18 -20 -6 -10 -12[5,3,2-]
1012510n­154 8 14 2 -16 -18 6 -20 -10 -12[5,21,2-]
1012610n­174 8 -14 2 -16 -18 -6 -20 -10 -12[41,3,2-]
1012710n­164 8 -14 2 16 18 -6 20 10 12[41,21,2-]
1012810n­224 8 -14 2 -16 -18 -20 -6 -12 -10[32,3,2-]
1012910n­184 8 14 2 -16 -18 6 -20 -12 -10[32,21,-2]
1013010n­204 8 -14 2 -16 -18 -6 -20 -12 -10[311,3,2-]
1013110n­194 8 -14 2 16 18 -6 20 12 10[311,21,2-]
1013210n­134 8 -12 2 -16 -6 -20 -18 -10 -14[23,3,2-]
1013310n44 8 12 2 -14 -18 6 -20 -10 -16[23,21,2-]
1013410n64 8 -12 2 -14 -18 -6 -20 -10 -16[221,3,2-]
1013510n54 8 -12 2 14 18 -6 20 10 16[221,21,2-]
1013610n34 8 10 -14 2 -18 -6 -20 -12 -16[22,22,2-]
1013710n24 8 10 -14 2 -16 -18 -6 -20 -12[22,211,2-]
1013810n14 8 10 -14 2 16 18 -6 20 12[211,211,2-]
1013910n­274 10 -14 -16 2 -18 -20 -6 -8 -12[4,3,3-]
1014010n­294 10 -14 -16 2 18 20 -8 -6 12[4,3,21-]
1014110n­254 10 -14 -16 2 18 -8 -6 20 12[4,21,21-]
1014210n­304 10 -14 -16 2 -18 -20 -8 -6 -12[31,3,3-]
1014310n­264 10 -14 -16 2 -18 -8 -6 -20 -12[31,3,21-]
1014410n­284 10 14 16 2 -18 -20 8 6 -12[31,21,21-]
1014510n­144 8 -12 -18 2 -16 -20 -6 -10 -14[22,3,3-]
1014610n­234 8 -18 -12 2 -16 -20 -6 -10 -14[22,21,21-]
1014710n­244 10 -14 12 2 16 18 -20 8 -6[211,3,21-]
1014810n­124 8 -12 2 -16 -6 -18 -20 -10 -14[(3,2)(3,2-)]
1014910n­114 8 -12 2 16 -6 18 20 10 14[(3,2)(21,2-)]
1015010n94 8 -12 2 -16 -6 -18 -10 -20 -14[(21,2)(3,2-)]
1015110n84 8 -12 2 16 -6 18 10 20 14[(21,2)(21,2-)]
1015210n­366 8 12 2 -16 4 -18 -20 -10 -14[(3,2)-(3,2)]
1015310n­104 8 12 2 -16 6 -18 -20 -10 -14[(3,2)-(21,2)]
1015410n74 8 12 2 -16 6 -18 -10 -20 -14[(21,2)-(21,2)]
1015510n­396 10 14 16 18 4 -20 2 8 -12[-3:2:2]
1015610n­324 12 16 -14 18 2 -8 20 10 6[-3:2:20]
1015710n­426 -10 -18 14 -2 -16 20 8 -4 12[-3:20:20]
1015810n­416 -10 -16 14 -2 -18 8 20 -4 -12[-30:2:2]
1015910n­346 8 10 14 16 -18 -20 2 4 -12[-30:2:20]
1016010n­334 12 -16 -14 -18 2 -8 -20 -10 -6[-30:20:20]
1016110n­314 12 -16 14 -18 2 8 -20 -10 -6[3:-20:-20]
1016210n­406 10 14 18 16 4 -20 2 8 -12[-30:-20:-20]
1016310n­356 8 10 14 16 -20 -18 2 4 -12[8*-30]
1016410n­386 -10 -12 14 -18 -16 20 -2 -4 -8[8*2:-20]
1016510n­376 8 14 18 16 4 -20 10 2 -12[8*2:.-20]

Higher

Table of prime links

Eight or fewer crossings

NamePictureAlexander–
Briggs–
Rolfsen
Dowker–
Thistlethwaite
Dowker
notation
Conway
notation
Unlink0 - - -
Hopf link2L2a1 - [2]
Solomon's
knot
4L4a1 - [4]
Whitehead
link
5L5a1 - [212]
L6a16L6a1 - -
L6a26L6a2 - -
L6a36L6a3 - -
Borromean
rings
6L6a4 - [.1]
L6a56L6a5 - -
L6n16L6n1 - -
L7a17L7a1 - -
L7a27L7a2 - -
L7a37L7a3 - -
L7a47L7a4 - -
L7a57L7a5 - -
L7a67L7a6 - -
L7a77L7a7 - -
L7n17L7n1 - -
L7n27L7n2(6,-8|-10,12,-14,2,-4)| - |}

Higher

PictureAlexander–
Briggs–
Rolfsen
Dowker–
Thistlethwaite
Dowker
notation
Conway
notation
8L8a14 - -
- L10a140 - [.3:30]

See also

External links