9 12

9_11

9_13

Visit 9 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 12's page at Knotilus! Visit 9 12's page at the original Knot Atlas! 9 12 Quick Notes

9 12 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code	4 10 16 14 2 18 8 6 12
Conway Notation	[4212]

Minimum Braid Representative: Length is 10, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,6\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 8.83664 A-Polynomial See Data:9 12/A-polynomial

[edit Notes for 9 12's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Rasmussen s-Invariant -2

[edit Notes for 9 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^2+9 t-13+9 t^{-1} -2 t^{-2} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^4+z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 35, -2 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q-2+4 q^{-1} -5 q^{-2} +6 q^{-3} -6 q^{-4} +5 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^8+2 z^2 a^6+2 a^6-z^4 a^4-z^2 a^4-a^4-z^4 a^2-z^2 a^2+z^2+1}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+2 z^7 a^7-5 z^5 a^7+3 z^3 a^7-z a^7+z^8 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6+4 z^7 a^5-11 z^5 a^5+13 z^3 a^5-4 z a^5+z^8 a^4-z^4 a^4+3 z^2 a^4-a^4+2 z^7 a^3-3 z^5 a^3+4 z^3 a^3-2 z a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{26}-q^{24}+q^{22}+q^{18}+2 q^{16}-q^{14}-q^{10}+q^6-q^4+2 q^2+ q^{-4} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{128}-q^{126}+2 q^{124}-3 q^{122}+2 q^{120}-2 q^{118}-2 q^{116}+7 q^{114}-10 q^{112}+10 q^{110}-10 q^{108}+4 q^{106}+4 q^{104}-15 q^{102}+20 q^{100}-20 q^{98}+14 q^{96}-q^{94}-12 q^{92}+19 q^{90}-18 q^{88}+15 q^{86}-3 q^{84}-10 q^{82}+14 q^{80}-11 q^{78}+2 q^{76}+10 q^{74}-16 q^{72}+18 q^{70}-8 q^{68}-4 q^{66}+17 q^{64}-26 q^{62}+29 q^{60}-21 q^{58}+6 q^{56}+11 q^{54}-23 q^{52}+30 q^{50}-25 q^{48}+13 q^{46}-13 q^{42}+16 q^{40}-14 q^{38}+3 q^{36}+8 q^{34}-13 q^{32}+10 q^{30}-2 q^{28}-9 q^{26}+17 q^{24}-19 q^{22}+15 q^{20}-6 q^{18}-5 q^{16}+14 q^{14}-16 q^{12}+17 q^{10}-10 q^8+5 q^6+q^4-7 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }

Further Quantum Invariants

Further quantum knot invariants for 9_12.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{17}+q^{15}-q^{13}+2 q^{11}-q^9+q^5-q^3+2 q- q^{-1} + q^{-3} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-q^{46}-q^{44}+3 q^{42}-2 q^{40}-4 q^{38}+5 q^{36}+q^{34}-6 q^{32}+5 q^{30}+3 q^{28}-7 q^{26}+2 q^{24}+3 q^{22}-3 q^{20}-2 q^{18}+2 q^{16}+4 q^{14}-5 q^{12}+7 q^8-5 q^6-2 q^4+7 q^2-2-2 q^{-2} +3 q^{-4} - q^{-6} - q^{-8} + q^{-10} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{93}+q^{91}+q^{89}-q^{87}-2 q^{85}+2 q^{83}+5 q^{81}-2 q^{79}-8 q^{77}+10 q^{73}+3 q^{71}-12 q^{69}-10 q^{67}+12 q^{65}+15 q^{63}-6 q^{61}-19 q^{59}+3 q^{57}+21 q^{55}+2 q^{53}-21 q^{51}-7 q^{49}+19 q^{47}+7 q^{45}-14 q^{43}-9 q^{41}+9 q^{39}+10 q^{37}-5 q^{35}-10 q^{33}-3 q^{31}+11 q^{29}+10 q^{27}-10 q^{25}-16 q^{23}+9 q^{21}+20 q^{19}-5 q^{17}-21 q^{15}+q^{13}+20 q^{11}+4 q^9-15 q^7-5 q^5+11 q^3+5 q-5 q^{-1} -4 q^{-3} +3 q^{-5} +2 q^{-7} -2 q^{-9} +2 q^{-13} - q^{-17} - q^{-19} + q^{-21} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{152}-q^{150}-q^{148}+q^{146}+2 q^{142}-4 q^{140}-3 q^{138}+4 q^{136}+3 q^{134}+8 q^{132}-8 q^{130}-13 q^{128}+2 q^{126}+8 q^{124}+23 q^{122}-4 q^{120}-25 q^{118}-17 q^{116}-4 q^{114}+40 q^{112}+23 q^{110}-12 q^{108}-36 q^{106}-41 q^{104}+28 q^{102}+50 q^{100}+33 q^{98}-24 q^{96}-77 q^{94}-14 q^{92}+44 q^{90}+69 q^{88}+12 q^{86}-80 q^{84}-49 q^{82}+16 q^{80}+74 q^{78}+38 q^{76}-56 q^{74}-51 q^{72}-5 q^{70}+52 q^{68}+39 q^{66}-25 q^{64}-39 q^{62}-15 q^{60}+29 q^{58}+32 q^{56}+3 q^{54}-30 q^{52}-29 q^{50}+31 q^{46}+43 q^{44}-17 q^{42}-50 q^{40}-36 q^{38}+24 q^{36}+79 q^{34}+13 q^{32}-50 q^{30}-68 q^{28}-5 q^{26}+84 q^{24}+42 q^{22}-18 q^{20}-67 q^{18}-38 q^{16}+48 q^{14}+42 q^{12}+18 q^{10}-33 q^8-40 q^6+10 q^4+14 q^2+24- q^{-2} -18 q^{-4} -2 q^{-6} -5 q^{-8} +10 q^{-10} +4 q^{-12} -3 q^{-14} +3 q^{-16} -6 q^{-18} +4 q^{-26} - q^{-28} - q^{-32} - q^{-34} + q^{-36} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{225}+q^{223}+q^{221}-q^{219}+2 q^{211}+2 q^{209}-4 q^{207}-6 q^{205}-q^{203}+4 q^{201}+9 q^{199}+8 q^{197}-4 q^{195}-19 q^{193}-17 q^{191}+3 q^{189}+23 q^{187}+33 q^{185}+13 q^{183}-24 q^{181}-50 q^{179}-36 q^{177}+12 q^{175}+56 q^{173}+66 q^{171}+23 q^{169}-47 q^{167}-91 q^{165}-71 q^{163}+6 q^{161}+91 q^{159}+119 q^{157}+58 q^{155}-56 q^{153}-149 q^{151}-138 q^{149}-4 q^{147}+147 q^{145}+202 q^{143}+99 q^{141}-106 q^{139}-247 q^{137}-186 q^{135}+40 q^{133}+245 q^{131}+256 q^{129}+44 q^{127}-220 q^{125}-299 q^{123}-113 q^{121}+170 q^{119}+300 q^{117}+164 q^{115}-109 q^{113}-278 q^{111}-188 q^{109}+65 q^{107}+231 q^{105}+181 q^{103}-19 q^{101}-182 q^{99}-164 q^{97}-3 q^{95}+138 q^{93}+134 q^{91}+17 q^{89}-96 q^{87}-114 q^{85}-35 q^{83}+68 q^{81}+106 q^{79}+51 q^{77}-34 q^{75}-103 q^{73}-93 q^{71}+q^{69}+108 q^{67}+140 q^{65}+48 q^{63}-107 q^{61}-193 q^{59}-114 q^{57}+89 q^{55}+238 q^{53}+187 q^{51}-47 q^{49}-263 q^{47}-256 q^{45}-19 q^{43}+249 q^{41}+307 q^{39}+93 q^{37}-201 q^{35}-316 q^{33}-160 q^{31}+122 q^{29}+287 q^{27}+207 q^{25}-35 q^{23}-224 q^{21}-210 q^{19}-36 q^{17}+139 q^{15}+184 q^{13}+84 q^{11}-65 q^9-135 q^7-90 q^5+7 q^3+80 q+81 q^{-1} +26 q^{-3} -39 q^{-5} -57 q^{-7} -32 q^{-9} +8 q^{-11} +33 q^{-13} +28 q^{-15} +7 q^{-17} -15 q^{-19} -20 q^{-21} -8 q^{-23} +3 q^{-25} +9 q^{-27} +9 q^{-29} +4 q^{-31} -5 q^{-33} -6 q^{-35} -2 q^{-37} -2 q^{-39} +2 q^{-41} +4 q^{-43} + q^{-45} - q^{-47} - q^{-51} - q^{-53} + q^{-55} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{26}-q^{24}+q^{22}+q^{18}+2 q^{16}-q^{14}-q^{10}+q^6-q^4+2 q^2+ q^{-4} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{68}-2 q^{66}+4 q^{64}-8 q^{62}+17 q^{60}-24 q^{58}+32 q^{56}-48 q^{54}+57 q^{52}-64 q^{50}+62 q^{48}-58 q^{46}+46 q^{44}-18 q^{42}-6 q^{40}+38 q^{38}-65 q^{36}+90 q^{34}-112 q^{32}+116 q^{30}-122 q^{28}+110 q^{26}-90 q^{24}+68 q^{22}-37 q^{20}+10 q^{18}+20 q^{16}-36 q^{14}+47 q^{12}-54 q^{10}+56 q^8-48 q^6+42 q^4-36 q^2+30-20 q^{-2} +15 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}+q^{64}-2 q^{60}-q^{58}+q^{56}-q^{54}-3 q^{52}-q^{50}+4 q^{48}+3 q^{46}-2 q^{44}+4 q^{40}-5 q^{36}-3 q^{34}+2 q^{32}-2 q^{28}+2 q^{26}+q^{24}-q^{22}+3 q^{20}+2 q^{18}-3 q^{16}-q^{14}+4 q^{12}+q^{10}-4 q^8+6 q^4-3+ q^{-2} +2 q^{-4} - q^{-8} + q^{-12} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{54}-q^{52}+q^{48}-3 q^{46}+q^{44}+2 q^{42}-5 q^{40}+2 q^{38}+4 q^{36}-6 q^{34}+q^{32}+4 q^{30}-3 q^{28}+3 q^{24}+q^{22}-q^{20}-q^{18}+4 q^{16}-2 q^{14}-4 q^{12}+6 q^{10}-q^8-5 q^6+5 q^4+q^2-2+3 q^{-2} + q^{-4} - q^{-6} + q^{-8} }
1,0,0	$-q^{35}-q^{33}-q^{31}+q^{29}+2q^{25}+q^{23}+2q^{21}-q^{19}-q^{15}-q^{13}+q^{7}-q^{5}+2q^{3}+q^{-1}+q^{-5}$

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{54}+q^{52}-2 q^{50}+3 q^{48}-5 q^{46}+5 q^{44}-6 q^{42}+5 q^{40}-4 q^{38}+4 q^{36}-q^{32}+6 q^{30}-7 q^{28}+10 q^{26}-11 q^{24}+11 q^{22}-11 q^{20}+7 q^{18}-6 q^{16}+2 q^{14}-2 q^{10}+5 q^8-5 q^6+7 q^4-5 q^2+4-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{88}-q^{84}-q^{82}+q^{80}+2 q^{78}-q^{76}-4 q^{74}-q^{72}+4 q^{70}+4 q^{68}-3 q^{66}-6 q^{64}+6 q^{60}+4 q^{58}-5 q^{56}-5 q^{54}+q^{52}+5 q^{50}-4 q^{46}-q^{44}+4 q^{42}+2 q^{40}-3 q^{38}-2 q^{36}+3 q^{34}+4 q^{32}-2 q^{30}-4 q^{28}+q^{26}+5 q^{24}-5 q^{20}-2 q^{18}+5 q^{16}+5 q^{14}-3 q^{12}-6 q^{10}-q^8+6 q^6+3 q^4-2 q^2-3+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{128}-q^{126}+2 q^{124}-3 q^{122}+2 q^{120}-2 q^{118}-2 q^{116}+7 q^{114}-10 q^{112}+10 q^{110}-10 q^{108}+4 q^{106}+4 q^{104}-15 q^{102}+20 q^{100}-20 q^{98}+14 q^{96}-q^{94}-12 q^{92}+19 q^{90}-18 q^{88}+15 q^{86}-3 q^{84}-10 q^{82}+14 q^{80}-11 q^{78}+2 q^{76}+10 q^{74}-16 q^{72}+18 q^{70}-8 q^{68}-4 q^{66}+17 q^{64}-26 q^{62}+29 q^{60}-21 q^{58}+6 q^{56}+11 q^{54}-23 q^{52}+30 q^{50}-25 q^{48}+13 q^{46}-13 q^{42}+16 q^{40}-14 q^{38}+3 q^{36}+8 q^{34}-13 q^{32}+10 q^{30}-2 q^{28}-9 q^{26}+17 q^{24}-19 q^{22}+15 q^{20}-6 q^{18}-5 q^{16}+14 q^{14}-16 q^{12}+17 q^{10}-10 q^8+5 q^6+q^4-7 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 12"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^2+9 t-13+9 t^{-1} -2 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^4+z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 35, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q-2+4 q^{-1} -5 q^{-2} +6 q^{-3} -6 q^{-4} +5 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8} }

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^8+2 z^2 a^6+2 a^6-z^4 a^4-z^2 a^4-a^4-z^4 a^2-z^2 a^2+z^2+1}

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+2 z^7 a^7-5 z^5 a^7+3 z^3 a^7-z a^7+z^8 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6+4 z^7 a^5-11 z^5 a^5+13 z^3 a^5-4 z a^5+z^8 a^4-z^4 a^4+3 z^2 a^4-a^4+2 z^7 a^3-3 z^5 a^3+4 z^3 a^3-2 z a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n84, ...}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {K11n15, ...}

Vassiliev invariants

V2 and V3:

(1, -3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} $-24$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{302}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{58}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -96} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -400} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -32} $-120$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{32}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1208}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{232}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{49471}{30}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{5942}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{49622}{45}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1793}{18}} ${\frac {4831}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -2 is the signature of 9 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                 1   -1 -1               3 1   2 -3             3 2     -1 -5           3 2       1 -7         3 3         0 -9       2 3           -1 -11     1 3             2 -13   1 2               -1 -15   1                 1 -17 1                   -1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	$J_{n}$
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-2 q^3+5 q-7+14 q^{-2} -16 q^{-3} -3 q^{-4} +26 q^{-5} -23 q^{-6} -8 q^{-7} +35 q^{-8} -25 q^{-9} -12 q^{-10} +34 q^{-11} -19 q^{-12} -13 q^{-13} +25 q^{-14} -9 q^{-15} -11 q^{-16} +14 q^{-17} -2 q^{-18} -7 q^{-19} +5 q^{-20} -2 q^{-22} + q^{-23} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^9-2 q^8+q^6+3 q^5-4 q^4-2 q^3+5 q^2+4 q-11-3 q^{-1} +15 q^{-2} +10 q^{-3} -27 q^{-4} -13 q^{-5} +34 q^{-6} +26 q^{-7} -46 q^{-8} -35 q^{-9} +50 q^{-10} +51 q^{-11} -57 q^{-12} -60 q^{-13} +56 q^{-14} +71 q^{-15} -56 q^{-16} -74 q^{-17} +49 q^{-18} +76 q^{-19} -41 q^{-20} -75 q^{-21} +31 q^{-22} +71 q^{-23} -20 q^{-24} -63 q^{-25} +5 q^{-26} +57 q^{-27} +3 q^{-28} -44 q^{-29} -13 q^{-30} +35 q^{-31} +16 q^{-32} -23 q^{-33} -16 q^{-34} +13 q^{-35} +14 q^{-36} -8 q^{-37} -9 q^{-38} +3 q^{-39} +6 q^{-40} -2 q^{-41} -2 q^{-42} +2 q^{-44} - q^{-45} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{16}-2 q^{15}+q^{13}-q^{12}+6 q^{11}-6 q^{10}+q^8-7 q^7+15 q^6-12 q^5+7 q^4+7 q^3-22 q^2+18 q-28+24 q^{-1} +32 q^{-2} -32 q^{-3} +14 q^{-4} -78 q^{-5} +31 q^{-6} +83 q^{-7} -8 q^{-8} +20 q^{-9} -164 q^{-10} +2 q^{-11} +132 q^{-12} +52 q^{-13} +62 q^{-14} -253 q^{-15} -61 q^{-16} +150 q^{-17} +115 q^{-18} +128 q^{-19} -308 q^{-20} -121 q^{-21} +136 q^{-22} +148 q^{-23} +188 q^{-24} -320 q^{-25} -152 q^{-26} +107 q^{-27} +147 q^{-28} +221 q^{-29} -291 q^{-30} -155 q^{-31} +63 q^{-32} +123 q^{-33} +235 q^{-34} -227 q^{-35} -142 q^{-36} +6 q^{-37} +77 q^{-38} +230 q^{-39} -133 q^{-40} -106 q^{-41} -52 q^{-42} +12 q^{-43} +199 q^{-44} -41 q^{-45} -49 q^{-46} -77 q^{-47} -46 q^{-48} +136 q^{-49} +12 q^{-50} +8 q^{-51} -60 q^{-52} -68 q^{-53} +67 q^{-54} +17 q^{-55} +32 q^{-56} -25 q^{-57} -51 q^{-58} +23 q^{-59} +4 q^{-60} +24 q^{-61} -4 q^{-62} -24 q^{-63} +8 q^{-64} -2 q^{-65} +9 q^{-66} + q^{-67} -8 q^{-68} +3 q^{-69} - q^{-70} +2 q^{-71} -2 q^{-73} + q^{-74} }
5	$q^{25}-2q^{24}+q^{22}-q^{21}+2q^{20}+4q^{19}-4q^{18}-4q^{17}+q^{16}-5q^{15}+3q^{14}+13q^{13}+q^{12}-4q^{11}-5q^{10}-16q^{9}-9q^{8}+18q^{7}+23q^{6}+17q^{5}-41q^{3}-49q^{2}-7q+41+82q^{-1}+55q^{-2}-42q^{-3}-122q^{-4}-104q^{-5}-4q^{-6}+152q^{-7}+204q^{-8}+58q^{-9}-167q^{-10}-279q^{-11}-178q^{-12}+138q^{-13}+393q^{-14}+300q^{-15}-87q^{-16}-444q^{-17}-460q^{-18}-18q^{-19}+508q^{-20}+594q^{-21}+127q^{-22}-502q^{-23}-728q^{-24}-255q^{-25}+501q^{-26}+810q^{-27}+361q^{-28}-451q^{-29}-877q^{-30}-458q^{-31}+422q^{-32}+896q^{-33}+516q^{-34}-359q^{-35}-909q^{-36}-565q^{-37}+328q^{-38}+886q^{-39}+585q^{-40}-274q^{-41}-854q^{-42}-603q^{-43}+225q^{-44}+807q^{-45}+603q^{-46}-161q^{-47}-737q^{-48}-599q^{-49}+84q^{-50}+646q^{-51}+585q^{-52}+2q^{-53}-537q^{-54}-549q^{-55}-82q^{-56}+393q^{-57}+495q^{-58}+171q^{-59}-264q^{-60}-413q^{-61}-212q^{-62}+110q^{-63}+309q^{-64}+250q^{-65}-201q^{-67}-223q^{-68}-95q^{-69}+83q^{-70}+189q^{-71}+141q^{-72}+4q^{-73}-120q^{-74}-150q^{-75}-68q^{-76}+55q^{-77}+130q^{-78}+97q^{-79}-6q^{-80}-89q^{-81}-96q^{-82}-30q^{-83}+53q^{-84}+77q^{-85}+38q^{-86}-19q^{-87}-53q^{-88}-40q^{-89}+9q^{-90}+29q^{-91}+24q^{-92}+7q^{-93}-16q^{-94}-20q^{-95}-q^{-96}+9q^{-97}+4q^{-98}+5q^{-99}-q^{-100}-8q^{-101}+4q^{-103}-q^{-104}+q^{-106}-2q^{-107}+2q^{-109}-q^{-110}$
6	$q^{36}-2q^{35}+q^{33}-q^{32}+2q^{31}+6q^{29}-8q^{28}-4q^{27}+3q^{26}-6q^{25}+4q^{24}+4q^{23}+23q^{22}-14q^{21}-10q^{20}+q^{19}-23q^{18}-5q^{17}+7q^{16}+63q^{15}-8q^{14}-6q^{13}+q^{12}-66q^{11}-48q^{10}-9q^{9}+128q^{8}+34q^{7}+41q^{6}+30q^{5}-140q^{4}-167q^{3}-103q^{2}+179q+123+203q^{-1}+184q^{-2}-185q^{-3}-384q^{-4}-388q^{-5}+74q^{-6}+164q^{-7}+514q^{-8}+622q^{-9}+3q^{-10}-566q^{-11}-904q^{-12}-390q^{-13}-102q^{-14}+801q^{-15}+1362q^{-16}+649q^{-17}-403q^{-18}-1420q^{-19}-1200q^{-20}-893q^{-21}+729q^{-22}+2109q^{-23}+1675q^{-24}+284q^{-25}-1581q^{-26}-2006q^{-27}-2038q^{-28}+171q^{-29}+2492q^{-30}+2666q^{-31}+1258q^{-32}-1295q^{-33}-2446q^{-34}-3078q^{-35}-598q^{-36}+2439q^{-37}+3266q^{-38}+2081q^{-39}-819q^{-40}-2475q^{-41}-3703q^{-42}-1212q^{-43}+2177q^{-44}+3457q^{-45}+2535q^{-46}-430q^{-47}-2300q^{-48}-3934q^{-49}-1547q^{-50}+1910q^{-51}+3419q^{-52}+2693q^{-53}-174q^{-54}-2078q^{-55}-3930q^{-56}-1709q^{-57}+1630q^{-58}+3254q^{-59}+2709q^{-60}+79q^{-61}-1770q^{-62}-3764q^{-63}-1836q^{-64}+1206q^{-65}+2904q^{-66}+2635q^{-67}+435q^{-68}-1252q^{-69}-3366q^{-70}-1943q^{-71}+568q^{-72}+2263q^{-73}+2379q^{-74}+848q^{-75}-500q^{-76}-2642q^{-77}-1884q^{-78}-154q^{-79}+1349q^{-80}+1810q^{-81}+1092q^{-82}+303q^{-83}-1635q^{-84}-1485q^{-85}-652q^{-86}+403q^{-87}+959q^{-88}+938q^{-89}+810q^{-90}-626q^{-91}-773q^{-92}-670q^{-93}-204q^{-94}+121q^{-95}+428q^{-96}+794q^{-97}+6q^{-98}-73q^{-99}-295q^{-100}-283q^{-101}-336q^{-102}-82q^{-103}+412q^{-104}+124q^{-105}+265q^{-106}+83q^{-107}-43q^{-108}-333q^{-109}-281q^{-110}+63q^{-111}-25q^{-112}+223q^{-113}+196q^{-114}+145q^{-115}-138q^{-116}-201q^{-117}-49q^{-118}-120q^{-119}+69q^{-120}+115q^{-121}+152q^{-122}-17q^{-123}-75q^{-124}-19q^{-125}-94q^{-126}-6q^{-127}+28q^{-128}+82q^{-129}+6q^{-130}-20q^{-131}+10q^{-132}-40q^{-133}-12q^{-134}-q^{-135}+31q^{-136}+q^{-137}-8q^{-138}+11q^{-139}-11q^{-140}-4q^{-141}-3q^{-142}+10q^{-143}-q^{-144}-5q^{-145}+5q^{-146}-2q^{-147}-q^{-149}+2q^{-150}-2q^{-152}+q^{-153}$
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{49}-2 q^{48}+q^{46}-q^{45}+2 q^{44}+2 q^{42}+2 q^{41}-8 q^{40}-2 q^{39}+2 q^{38}-5 q^{37}+6 q^{36}+2 q^{35}+10 q^{34}+14 q^{33}-20 q^{32}-8 q^{31}-3 q^{30}-18 q^{29}+5 q^{28}+q^{27}+27 q^{26}+47 q^{25}-22 q^{24}-15 q^{23}-17 q^{22}-52 q^{21}-3 q^{20}-13 q^{19}+47 q^{18}+115 q^{17}+2 q^{16}-10 q^{15}-54 q^{14}-129 q^{13}-46 q^{12}-45 q^{11}+87 q^{10}+245 q^9+105 q^8+56 q^7-107 q^6-328 q^5-232 q^4-190 q^3+107 q^2+497 q+437+394 q^{-1} -21 q^{-2} -633 q^{-3} -751 q^{-4} -785 q^{-5} -213 q^{-6} +719 q^{-7} +1108 q^{-8} +1367 q^{-9} +734 q^{-10} -604 q^{-11} -1499 q^{-12} -2163 q^{-13} -1532 q^{-14} +211 q^{-15} +1679 q^{-16} +3035 q^{-17} +2740 q^{-18} +666 q^{-19} -1634 q^{-20} -3948 q^{-21} -4153 q^{-22} -1911 q^{-23} +1075 q^{-24} +4556 q^{-25} +5758 q^{-26} +3676 q^{-27} -81 q^{-28} -4912 q^{-29} -7247 q^{-30} -5570 q^{-31} -1417 q^{-32} +4701 q^{-33} +8508 q^{-34} +7619 q^{-35} +3201 q^{-36} -4138 q^{-37} -9366 q^{-38} -9411 q^{-39} -5108 q^{-40} +3157 q^{-41} +9801 q^{-42} +10930 q^{-43} +6913 q^{-44} -2041 q^{-45} -9850 q^{-46} -11991 q^{-47} -8469 q^{-48} +875 q^{-49} +9628 q^{-50} +12691 q^{-51} +9659 q^{-52} +131 q^{-53} -9240 q^{-54} -13006 q^{-55} -10516 q^{-56} -995 q^{-57} +8840 q^{-58} +13141 q^{-59} +11041 q^{-60} +1579 q^{-61} -8434 q^{-62} -13063 q^{-63} -11373 q^{-64} -2057 q^{-65} +8099 q^{-66} +12985 q^{-67} +11532 q^{-68} +2363 q^{-69} -7775 q^{-70} -12798 q^{-71} -11617 q^{-72} -2689 q^{-73} +7427 q^{-74} +12593 q^{-75} +11682 q^{-76} +3005 q^{-77} -7008 q^{-78} -12276 q^{-79} -11681 q^{-80} -3415 q^{-81} +6397 q^{-82} +11827 q^{-83} +11667 q^{-84} +3922 q^{-85} -5618 q^{-86} -11181 q^{-87} -11530 q^{-88} -4511 q^{-89} +4590 q^{-90} +10267 q^{-91} +11255 q^{-92} +5150 q^{-93} -3337 q^{-94} -9083 q^{-95} -10777 q^{-96} -5738 q^{-97} +1963 q^{-98} +7609 q^{-99} +9957 q^{-100} +6175 q^{-101} -463 q^{-102} -5890 q^{-103} -8898 q^{-104} -6365 q^{-105} -848 q^{-106} +4064 q^{-107} +7429 q^{-108} +6152 q^{-109} +2011 q^{-110} -2213 q^{-111} -5819 q^{-112} -5594 q^{-113} -2696 q^{-114} +633 q^{-115} +4038 q^{-116} +4596 q^{-117} +2962 q^{-118} +657 q^{-119} -2385 q^{-120} -3432 q^{-121} -2731 q^{-122} -1386 q^{-123} +993 q^{-124} +2130 q^{-125} +2124 q^{-126} +1672 q^{-127} -4 q^{-128} -999 q^{-129} -1338 q^{-130} -1489 q^{-131} -512 q^{-132} +135 q^{-133} +529 q^{-134} +1043 q^{-135} +632 q^{-136} +370 q^{-137} +107 q^{-138} -522 q^{-139} -459 q^{-140} -520 q^{-141} -489 q^{-142} +56 q^{-143} +163 q^{-144} +444 q^{-145} +608 q^{-146} +232 q^{-147} +118 q^{-148} -241 q^{-149} -546 q^{-150} -343 q^{-151} -278 q^{-152} +30 q^{-153} +369 q^{-154} +315 q^{-155} +358 q^{-156} +105 q^{-157} -224 q^{-158} -217 q^{-159} -294 q^{-160} -160 q^{-161} +65 q^{-162} +109 q^{-163} +245 q^{-164} +166 q^{-165} -26 q^{-166} -45 q^{-167} -134 q^{-168} -113 q^{-169} -24 q^{-170} -19 q^{-171} +92 q^{-172} +93 q^{-173} +11 q^{-174} +10 q^{-175} -42 q^{-176} -35 q^{-177} -10 q^{-178} -32 q^{-179} +18 q^{-180} +36 q^{-181} +6 q^{-182} +8 q^{-183} -14 q^{-184} -6 q^{-185} +6 q^{-186} -16 q^{-187} +10 q^{-189} +2 q^{-190} +3 q^{-191} -6 q^{-192} - q^{-193} +6 q^{-194} -4 q^{-195} -2 q^{-196} +2 q^{-197} + q^{-199} -2 q^{-200} +2 q^{-202} - q^{-203} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 12]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 16, 6, 17], X[11, 1, 12, 18], X[17, 13, 18, 12], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 6, 16, 7], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[9, 12]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4]
In[4]:=	DTCode[Knot[9, 12]]
Out[4]=	DTCode[4, 10, 16, 14, 2, 18, 8, 6, 12]
In[5]:=	br = BR[Knot[9, 12]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, 2, -3, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[9, 12]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 12]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 12]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 12]][t]
Out[10]=	2 9 2 -13 - -- + - + 9 t - 2 t 2 t t
In[11]:=	Conway[Knot[9, 12]][z]
Out[11]=	2 4 1 + z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 12], Knot[11, NonAlternating, 84]}
In[13]:=	{KnotDet[Knot[9, 12]], KnotSignature[Knot[9, 12]]}
Out[13]=	{35, -2}
In[14]:=	Jones[Knot[9, 12]][q]
Out[14]=	-8 2 3 5 6 6 5 4 -2 - q + -- - -- + -- - -- + -- - -- + - + q 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 12], Knot[11, NonAlternating, 15]}
In[16]:=	A2Invariant[Knot[9, 12]][q]
Out[16]=	-26 -24 -22 -18 2 -14 -10 -6 -4 2 4 -q - q + q + q + --- - q - q + q - q + -- + q 16 2 q q
In[17]:=	HOMFLYPT[Knot[9, 12]][a, z]
Out[17]=	4 6 8 2 2 2 4 2 6 2 2 4 4 4 1 - a + 2 a - a + z - a z - a z + 2 a z - a z - a z
In[18]:=	Kauffman[Knot[9, 12]][a, z]
Out[18]=	4 6 8 3 5 7 9 2 2 2 1 - a - 2 a - a - 2 a z - 4 a z - a z + a z - 2 z - 2 a z + 4 2 6 2 8 2 3 3 3 5 3 7 3 3 a z + 7 a z + 4 a z - 3 a z + 4 a z + 13 a z + 3 a z - 9 3 4 2 4 4 4 6 4 8 4 5 3 5 3 a z + z - a z - a z - 5 a z - 6 a z + 2 a z - 3 a z - 5 5 7 5 9 5 2 6 8 6 3 7 5 7 11 a z - 5 a z + a z + 2 a z + 2 a z + 2 a z + 4 a z + 7 7 4 8 6 8 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 12]], Vassiliev[3][Knot[9, 12]]}
Out[19]=	{1, -3}
In[20]:=	Kh[Knot[9, 12]][q, t]
Out[20]=	2 3 1 1 1 2 1 3 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 3 3 3 3 2 3 t 3 2 ----- + ----- + ----- + ----- + ---- + ---- + - + q t + q t 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[9, 12], 2][q]
Out[21]=	-23 2 5 7 2 14 11 9 25 13 19 -7 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + 22 20 19 18 17 16 15 14 13 12 q q q q q q q q q q 34 12 25 35 8 23 26 3 16 14 3 4 --- - --- - -- + -- - -- - -- + -- - -- - -- + -- + 5 q - 2 q + q 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q

See/edit the Rolfsen_Splice_Template.