List of prime knots

Last updated December 29, 2022

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

Name	Alexander– Briggs– Rolfsen	Dowker– Thistlethwaite	Dowker notation	Conway notation	crossinglist
Unknot	0₁	0a1	—	—	0
Trefoil knot	3₁	3a1	4 6 2	[3]	123:123
Figure-eight knot	4₁	4a1	4 6 8 2	[22]	1234:2143 1231\4324
Cinquefoil knot	5₁	5a2	6 8 10 2 4	[5]	12345:12345
Three-twist knot	5₂	5a1	4 8 10 2 6	[32]	12345:12543 1231\452354
Stevedore knot	6₁	6a3	4 8 12 10 2 6	[42]	123456:216543 1231\45632654
6₂ knot	6₂	6a2	4 8 10 12 2 6	[312]	123456:234165 1231\45632456
6₃ knot	6₃	6a1	4 8 10 2 12 6	[2112]	123456:236145 1231\45642356 1231\45236456

Seven crossings

Alexander– Briggs– Rolfsen	Dowker– Thistlethwaite	Dowker notation	Conway notation	crossinglist
7₁	7a7	8 10 12 14 2 4 6	[7]	1-7:1-7
7₂	7a4	4 10 14 12 2 8 6	[52]	1-7:127-3
7₃	7a5	6 10 12 14 2 4 8	[43]
7₄	7a6	6 10 12 14 4 2 8	[313]
7₅	7a3	4 10 12 14 2 8 6	[322]
7₆	7a2	4 8 12 2 14 6 10	[2212]
7₇	7a1	4 8 10 12 2 14 6	[21112]

Eight crossings

Alexander– Briggs– Rolfsen	Dowker– Thistlethwaite	Dowker notation	Conway notation
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]
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]
8₁₅	8a2	4 8 12 2 14 6 16 10	[21,21,2]
8₁₆	8a15	6 8 14 12 4 16 2 10	[.2.20]
8₁₇	8a14	6 8 12 14 4 16 2 10	[.2.2]
8₁₈	8a12	6 8 10 12 14 16 2 4	[8*]
8₁₉	8n3	4 8 -12 2 -14 -16 -6 -10	[3,3,2-]
8₂₀	8n1	4 8 -12 2 -14 -6 -16 -10	[3,21,2-]
8₂₁	8n2	4 8 -12 2 14 -6 16 10	[21,21,2-]

Nine crossings

Alexander– Briggs– Rolfsen	Dowker– Thistlethwaite	Dowker notation	Conway notation
9₁	9a41	10 12 14 16 18 2 4 6 8	[9]
9₂	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]
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]
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]
9₄₀	9a27	6 16 14 12 4 2 18 10 8	[9*]
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−]
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

Alexander– Briggs– Rolfsen	Dowker– Thistlethwaite	Dowker notation	Conway notation
10₁	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]
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]
10₅₉	10a2	4 8 10 14 2 18 6 20 12 16	[22,211,2]
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+]
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]
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]
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]
10₁₂₂	10a89	6 10 12 14 18 16 20 2 4 8	[9*.20]
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-]
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]
10₁₆₁^{[lower-alpha 1]}	10n31	4 12 -16 14 -18 2 8 -20 -10 -6	[3:-20:-20]
10₁₆₂^{[lower-alpha 2]}	10n40	6 10 14 18 16 4 -20 2 8 -12	[-30:-20:-20]
10₁₆₃^{[lower-alpha 3]}	10n35	6 8 10 14 16 -20 -18 2 4 -12	[8*-30]
10₁₆₄^{[lower-alpha 4]}	10n38	6 -10 -12 14 -18 -16 20 -2 -4 -8	[8*2:-20]
10₁₆₅^{[lower-alpha 5]}	10n37	6 8 14 18 16 4 -20 10 2 -12	[8*2:.-20]

Higher

Kinoshita-Terasaka & Conway knots Knudemutation.svg — Kinoshita–Terasaka & Conway knots

Conway knot 11n34
Kinoshita–Terasaka knot 11n42

Table of prime links

Seven or fewer crossings

Name	Alexander– Briggs– Rolfsen	Dowker– Thistlethwaite	Dowker notation	Conway notation
Unlink	0² ₁	—	—	—
Hopf link	2² ₁	L2a1	—	[2]
Solomon's knot	4² ₁	L4a1	—	[4]
Whitehead link	5² ₁	L5a1	—	[212]
L6a1	6² ₃	L6a1	—	—
L6a2	6² ₂	L6a2	—	—
L6a3	6² ₁	L6a3	—	—
Borromean rings	6³ ₂	L6a4	—	[.1]
L6a5	6³ ₁	L6a5	—	—
L6n1	6³ ₃	L6n1	—	—
L7a1	7² ₆	L7a1	—	—
L7a2	7² ₅	L7a2	—	—
L7a3	7² ₄	L7a3	—	—
L7a4	7² ₃	L7a4	—	—
L7a5	7² ₂	L7a5	—	—
L7a6	7² ₁	L7a6	—	—
L7a7	7³ ₁	L7a7	—	—
L7n1	7² ₇	L7n1	—	—
L7n2	7² ₈	L7n2	(6,-8\|-10,12,-14,2,-4)	—

Higher

Picture	Alexander– Briggs– Rolfsen	Dowker– Thistlethwaite	Dowker notation	Conway notation
	8² ₁	L8a14	—	—
	—	L10a140	—	[.3:30]

Notes

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

External links

" The Rolfsen Knot Table ", The Knot Atlas .
"KnotInfo", Indiana.edu.

Related Research Articles

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In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory. Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification, both in "enumeration" and "duplication removal".

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In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R³. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime. A torus knot is trivial if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.

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[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.

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v t e Knot theory (knots and links)
Hyperbolic	Figure-eight (4₁) Three-twist (5₂) Stevedore (6₁) 6₂ 6₃ Endless (7₄) Carrick mat (8₁₈) Perko pair (10₁₆₁) (−2,3,7) pretzel (12n242) Whitehead (5² ₁) Borromean rings (6³ ₂) L10a140
Satellite	Composite knots Granny Square Knot sum
Torus	Unknot (0₁) Trefoil (3₁) Cinquefoil (5₁) Septafoil (7₁) Unlink (0² ₁) Hopf (2² ₁) Solomon's (4² ₁)
Invariants	Alternating Arf invariant Bridge no. 2-bridge Brunnian Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability Unknotting no. and problem
Notation and operations	Alexander–Briggs notation Conway notation Dowker notation Flype Mutation Reidemeister move Skein relation Tabulation
Other	Alexander's theorem Berge Braid theory Conway sphere Complement Double torus Fibered Knot List of knots and links Ribbon Slice Sum Tait conjectures Twist Wild Writhe Surgery theory
Category Commons

List of prime knots

Contents

Table of prime knots

Six or fewer crossings

Seven crossings

Eight crossings

Nine crossings

Ten crossings

Higher

Table of prime links

Seven or fewer crossings

Higher

See also

Notes

External links

Related Research Articles