Conway's Knot Notation
A concise notation based on the concept of the tangle used by Conway (1967) to enumerate prime knots up to 11 crossings.
An algebraic knot containing no negative signs in its Conway knot notation is an alternating knot.
Conway's knot notation is implemented in the Wolfram Language as KnotData [knot, "ConwayNotation"]. Rolfsen (1976) gives a table that includes Conway's knot notation for prime knots on 10 or fewer crossings, as summarized in the table below.
0_1 ? 9_(15) 2322 10_(16) 4123 10_(66) 31,21,21 10_(116) 8*2:2
3_1 3 9_(16) 3,3,2+ 10_(17) 4114 10_(67) 22,3,21 10_(117) 8*2:20
4_1 22 9_(17) 21312 10_(18) 41122 10_(68) 211,3,3 10_(118) 8*2:.2
5_1 5 9_(18) 3222 10_(19) 41113 10_(69) 211,21,21 10_(119) 8*2:.20
5_2 32 9_(19) 23112 10_(20) 352 10_(70) 22,3,2+ 10_(120) 8*20::20
6_1 42 9_(20) 31212 10_(21) 3412 10_(71) 22,21,2+ 10_(121) 9*20
6_2 312 9_(21) 31122 10_(22) 3313 10_(72) 211,3,2+ 10_(122) 9*.20
6_3 2112 9_(22) 211,3,2 10_(23) 33112 10_(73) 211,21,2+ 10_(123) 10*
7_1 7 9_(23) 22122 10_(24) 3232 10_(74) 3,3,21+ 10_(124) 5,3,2-
7_2 52 9_(24) 3,21,2+ 10_(25) 32212 10_(75) 21,21,21+ 10_(125) 5,21,2-
7_3 43 9_(25) 22,21,2 10_(26) 32113 10_(76) 3,3,2++ 10_(126) 41,3,2-
7_4 313 9_(26) 311112 10_(27) 321112 10_(77) 3,21,2++ 10_(127) 41,21,2-
7_5 322 9_(27) 212112 10_(28) 31312 10_(78) 21,21,2++ 10_(128) 32,3,2-
7_6 2212 9_(28) 21,21,2+ 10_(29) 31222 10_(79) (3,2)(3,2) 10_(129) 32,21,-2
7_7 21112 9_(29) .2.20.2 10_(30) 312112 10_(80) (3,2)(21,2) 10_(130) 311,3,2-
8_1 62 9_(30) 211,21,2 10_(31) 31132 10_(81) (21,2)(21,2) 10_(131) 311,21,2-
8_2 512 9_(31) 2111112 10_(32) 311122 10_(82) .4.2 10_(132) 23,3,2-
8_3 44 9_(32) .21.20 10_(33) 311113 10_(83) .31.20 10_(133) 23,21,2-
8_4 413 9_(33) .21.2 10_(34) 2512 10_(84) .22.2 10_(134) 221,3,2-
8_5 3,3,2 9_(34) 8*20 10_(35) 2422 10_(85) .4.20 10_(135) 221,21,2-
8_6 332 9_(35) 3,3,3 10_(36) 24112 10_(86) .31.2 10_(136) 22,22,2-
8_7 4112 9_(36) 22,3,2 10_(37) 2332 10_(87) .22.20 10_(137) 22,211,2-
8_8 2312 9_(37) 3,21,21 10_(38) 23122 10_(88) .21.21 10_(138) 211,211,2-
8_9 3113 9_(38) .2.2.2 10_(39) 22312 10_(89) .21.210 10_(139) 4,3,3-
8_(10) 3,21,2 9_(39) 2:2:20 10_(40) 222112 10_(90) .3.2.2 10_(140) 4,3,21-
8_(11) 3212 9_(40) 9* 10_(41) 221212 10_(91) .3.2.20 10_(141) 4,21,21-
8_(12) 2222 9_(41) 20:20:20 10_(42) 2211112 10_(92) .21.2.20 10_(142) 31,3,3-
8_(13) 31112 9_(42) 22,3,2- 10_(43) 212212 10_(93) .3.20.2 10_(143) 31,3,21-
8_(14) 22112 9_(43) 211,3,2- 10_(44) 2121112 10_(94) .30.2.2 10_(144) 31,21,21-
8_(15) 21,21,2 9_(44) 22,21,2- 10_(45) 21111112 10_(95) .210.2.2 10_(145) 22,3,3-
8_(16) .2.20 9_(45) 211,21,2- 10_(46) 5,3,2 10_(96) .2.21.2 10_(146) 22,21,21-
8_(17) .2.2 9_(46) 3,3,21- 10_(47) 5,21,2 10_(97) .2.210.2 10_(147) 211,3,21-
8_(18) 8* 9_(47) 8*-20 10_(48) 41,3,2 10_(98) .2.2.2.20 10_(148) (3,2)(3,2-)
8_(19) 3,3,2- 9_(48) 21,21,21- 10_(49) 41,21,2 10_(99) .2.2.20.20 10_(149) (3,2)(21,2-)
8_(20) 3,21,2- 9_(49) -20:-20:-20 10_(50) 32,3,2 10_(100) 3:2:2 10_(150) (21,2)(3,2-)
8_(21) 21,21,2- 10_1 82 10_(51) 32,21,2 10_(101) 21:2:2 10_(151) (21,2)(21,2-)
9_1 9 10_2 712 10_(52) 311,3,2 10_(102) 3:2:20 10_(152) (3,2)-(3,2)
9_2 72 10_3 64 10_(53) 311,21,2 10_(103) 30:2:2 10_(153) (3,2)-(21,2)
9_3 63 10_4 613 10_(54) 23,3,2 10_(104) 3:20:20 10_(154) (21,2)-(21,2)
9_4 54 10_5 6112 10_(55) 23,21,2 10_(105) 21:20:20 10_(155) -3:2:2
9_5 513 10_6 532 10_(56) 221,3,2 10_(106) 30:2:20 10_(156) -3:2:20
9_6 522 10_7 5212 10_(57) 221,21,2 10_(107) 210:2:20 10_(157) -3:20:20
9_7 342 10_8 514 10_(58) 22,22,2 10_(108) 30:20:20 10_(158) -30:2:2
9_8 2412 10_9 5113 10_(59) 22,211,2 10_(109) 2.2.2.2 10_(159) -30:2:20
9_9 423 10_(10) 51112 10_(60) 211,211,2 10_(110) 2.2.2.20 10_(160) -30:20:20
9_(10) 333 10_(11) 433 10_(61) 4,3,3 10_(111) 2.2.20.2 10_(161) 3:-20:-20
9_(11) 4122 10_(12) 4312 10_(62) 4,3,21 10_(112) 8*3 10_(162) -30:-20:-20
9_(12) 4212 10_(13) 4222 10_(63) 4,21,21 10_(113) 8*21 10_(163) 8*-30
9_(13) 3213 10_(14) 42112 10_(64) 31,3,3 10_(114) 8*30 10_(164) 8*2:-20
9_(14) 41112 10_(15) 4132 10_(65) 31,3,21 10_(115) 8*20.20 10_(165) 8*2:.-20
Explore with Wolfram|Alpha
WolframAlpha
References
Conway, J. H. "An Enumeration of Knots and Links, and Some of Their Algebraic Properties." In Computational Problems in Abstract Algebra (Ed. J. Leech). Oxford, England: Pergamon Press, pp. 329-358, 1967.Rolfsen, D. "Table of Knots and Links." Appendix C in Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 280-287, 1976.Referenced on Wolfram|Alpha
Conway's Knot NotationCite this as:
Weisstein, Eric W. "Conway's Knot Notation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConwaysKnotNotation.html