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

Name	Picture	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]	1234:21431231\4324
Cinquefoil knot		51	5a2	6 8 10 2 4	[5]	12345:12345
Three-twist knot		52	5a1	4 8 10 2 6	[32]	12345:125431231\452354
Stevedore knot		61	6a3	4 8 12 10 2 6	[42]	123456:2165431231\45632654
62 knot		62	6a2	4 8 10 12 2 6	[312]	123456:2341651231\45632456
63 knot		63	6a1	4 8 10 2 12 6	[2112]	123456:2361451231\45642356 1231\45236456

Seven crossings

Picture	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

Picture	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

Picture	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

Picture	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	10n­31	4 12 -16 14 -18 2 8 -20 -10 -6	[3:-20:-20]
	10162	10n­40	6 10 14 18 16 4 -20 2 8 -12	[-30:-20:-20]
	10163	10n­35	6 8 10 14 16 -20 -18 2 4 -12	[8*-30]
	10164	10n­38	6 -10 -12 14 -18 -16 20 -2 -4 -8	[8*2:-20]
	10165	10n­37	6 8 14 18 16 4 -20 10 2 -12	[8*2:.-20]

Higher

• Conway knot 11n34
• Kinoshita–Terasaka knot 11n42

Table of prime links

Eight or fewer crossings

Name	Picture	Alexander– Briggs– Rolfsen	Dowker– Thistlethwaite	Dowker notation	Conway notation
Unlink		0	-	-	-
Hopf link		2	L2a1	-	[2]
Solomon's knot		4	L4a1	-	[4]
Whitehead link		5	L5a1	-	[212]
L6a1		6	L6a1	-	-
L6a2		6	L6a2	-	-
L6a3		6	L6a3	-	-
Borromean rings		6	L6a4	-	[.1]
L6a5		6	L6a5	-	-
L6n1		6	L6n1	-	-
L7a1		7	L7a1	-	-
L7a2		7	L7a2	-	-
L7a3		7	L7a3	-	-
L7a4		7	L7a4	-	-
L7a5		7	L7a5	-	-
L7a6		7	L7a6	-	-
L7a7		7	L7a7	-	-
L7n1		7	L7n1	-	-
L7n2		7	L7n2	(6,-8|-10,12,-14,2,-4)| - |} Higher Picture Alexander– Briggs– Rolfsen Dowker– Thistlethwaite Dowker notation Conway notation 8 L8a14 - - - L10a140 - [.3:30] See also List of knots List of mathematical knots and links Knot tabulation (−2,3,7) pretzel knot External links "KnotInfo", Indiana.edu.