9 18

9_17

9_19

Visit 9 18's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 18's page at Knotilus! Visit 9 18's page at the original Knot Atlas! 9 18 Quick Notes

9 18 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,14,6,15 X9,18,10,1 X17,6,18,7 X7,16,8,17 X15,8,16,9 X13,10,14,11 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4
Dowker-Thistlethwaite code	4 12 14 16 18 2 10 8 6
Conway Notation	[3222]

Minimum Braid Representative: Length is 11, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 10.0577 A-Polynomial See Data:9 18/A-polynomial

[edit Notes for 9 18's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math] Topological 4 genus [math]\displaystyle{ 2 }[/math] Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant -4

[edit Notes for 9 18's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t^2-10 t+13-10 t^{-1} +4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 41, -4 }
Jones polynomial	[math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +7 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}-a^{10}+z^4 a^8+z^2 a^8+2 z^4 a^6+4 z^2 a^6+a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-5 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-3 z^5 a^{11}+z^8 a^{10}+z^6 a^{10}-2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+4 z^7 a^9-5 z^5 a^9+z^3 a^9+z^8 a^8+2 z^6 a^8-2 z^4 a^8+2 z^7 a^7+z^5 a^7-4 z^3 a^7+2 z a^7+3 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{34}-2 q^{28}+q^{26}+q^{20}-q^{18}+2 q^{16}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{176}-q^{174}+3 q^{172}-4 q^{170}+3 q^{168}-2 q^{166}-2 q^{164}+10 q^{162}-15 q^{160}+18 q^{158}-15 q^{156}+5 q^{154}+9 q^{152}-26 q^{150}+36 q^{148}-37 q^{146}+23 q^{144}-2 q^{142}-24 q^{140}+37 q^{138}-38 q^{136}+27 q^{134}-6 q^{132}-16 q^{130}+24 q^{128}-23 q^{126}+5 q^{124}+16 q^{122}-29 q^{120}+31 q^{118}-16 q^{116}-7 q^{114}+34 q^{112}-51 q^{110}+54 q^{108}-40 q^{106}+9 q^{104}+23 q^{102}-48 q^{100}+58 q^{98}-47 q^{96}+23 q^{94}+4 q^{92}-26 q^{90}+32 q^{88}-25 q^{86}+4 q^{84}+16 q^{82}-23 q^{80}+20 q^{78}-2 q^{76}-18 q^{74}+35 q^{72}-36 q^{70}+29 q^{68}-12 q^{66}-11 q^{64}+29 q^{62}-34 q^{60}+33 q^{58}-18 q^{56}+6 q^{54}+6 q^{52}-14 q^{50}+16 q^{48}-13 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_18.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{23}+q^{21}-2 q^{19}+2 q^{17}-q^{15}+q^{11}-q^9+3 q^7-q^5+q^3 }[/math]
2	[math]\displaystyle{ q^{64}-q^{62}-q^{60}+4 q^{58}-2 q^{56}-6 q^{54}+8 q^{52}+q^{50}-10 q^{48}+8 q^{46}+4 q^{44}-11 q^{42}+2 q^{40}+5 q^{38}-4 q^{36}-3 q^{34}+3 q^{32}+5 q^{30}-8 q^{28}-q^{26}+11 q^{24}-8 q^{22}-4 q^{20}+11 q^{18}-3 q^{16}-3 q^{14}+5 q^{12}-q^8+q^6 }[/math]
3	[math]\displaystyle{ -q^{123}+q^{121}+q^{119}-q^{117}-3 q^{115}+2 q^{113}+7 q^{111}-q^{109}-13 q^{107}-3 q^{105}+18 q^{103}+11 q^{101}-22 q^{99}-21 q^{97}+21 q^{95}+31 q^{93}-14 q^{91}-39 q^{89}+9 q^{87}+42 q^{85}+q^{83}-42 q^{81}-10 q^{79}+38 q^{77}+15 q^{75}-30 q^{73}-18 q^{71}+19 q^{69}+21 q^{67}-8 q^{65}-23 q^{63}-7 q^{61}+22 q^{59}+18 q^{57}-20 q^{55}-33 q^{53}+17 q^{51}+41 q^{49}-9 q^{47}-45 q^{45}+2 q^{43}+41 q^{41}+6 q^{39}-35 q^{37}-11 q^{35}+27 q^{33}+10 q^{31}-14 q^{29}-9 q^{27}+10 q^{25}+7 q^{23}-4 q^{21}-3 q^{19}+3 q^{17}+2 q^{15}-q^{11}+q^9 }[/math]
4	[math]\displaystyle{ q^{200}-q^{198}-q^{196}+q^{194}+3 q^{190}-4 q^{188}-5 q^{186}+3 q^{184}+4 q^{182}+15 q^{180}-7 q^{178}-23 q^{176}-10 q^{174}+7 q^{172}+50 q^{170}+14 q^{168}-39 q^{166}-55 q^{164}-30 q^{162}+83 q^{160}+77 q^{158}-4 q^{156}-94 q^{154}-117 q^{152}+55 q^{150}+132 q^{148}+88 q^{146}-68 q^{144}-192 q^{142}-29 q^{140}+122 q^{138}+164 q^{136}+6 q^{134}-197 q^{132}-104 q^{130}+64 q^{128}+181 q^{126}+66 q^{124}-146 q^{122}-124 q^{120}+9 q^{118}+142 q^{116}+87 q^{114}-72 q^{112}-111 q^{110}-38 q^{108}+83 q^{106}+93 q^{104}+10 q^{102}-86 q^{100}-88 q^{98}+4 q^{96}+96 q^{94}+110 q^{92}-43 q^{90}-137 q^{88}-92 q^{86}+73 q^{84}+195 q^{82}+30 q^{80}-137 q^{78}-171 q^{76}+6 q^{74}+209 q^{72}+94 q^{70}-69 q^{68}-172 q^{66}-64 q^{64}+138 q^{62}+99 q^{60}+6 q^{58}-106 q^{56}-76 q^{54}+56 q^{52}+52 q^{50}+30 q^{48}-37 q^{46}-43 q^{44}+15 q^{42}+14 q^{40}+19 q^{38}-9 q^{36}-15 q^{34}+7 q^{32}+2 q^{30}+7 q^{28}-2 q^{26}-4 q^{24}+3 q^{22}+2 q^{18}-q^{14}+q^{12} }[/math]
5	[math]\displaystyle{ -q^{295}+q^{293}+q^{291}-q^{289}-q^{283}+2 q^{281}+4 q^{279}-3 q^{277}-7 q^{275}-4 q^{273}-q^{271}+11 q^{269}+20 q^{267}+8 q^{265}-20 q^{263}-39 q^{261}-29 q^{259}+14 q^{257}+67 q^{255}+74 q^{253}+13 q^{251}-86 q^{249}-137 q^{247}-76 q^{245}+73 q^{243}+199 q^{241}+182 q^{239}-5 q^{237}-239 q^{235}-308 q^{233}-121 q^{231}+209 q^{229}+422 q^{227}+305 q^{225}-100 q^{223}-487 q^{221}-502 q^{219}-79 q^{217}+462 q^{215}+665 q^{213}+314 q^{211}-353 q^{209}-771 q^{207}-532 q^{205}+180 q^{203}+772 q^{201}+718 q^{199}+25 q^{197}-709 q^{195}-827 q^{193}-204 q^{191}+583 q^{189}+846 q^{187}+349 q^{185}-443 q^{183}-803 q^{181}-430 q^{179}+309 q^{177}+714 q^{175}+452 q^{173}-186 q^{171}-605 q^{169}-448 q^{167}+91 q^{165}+500 q^{163}+427 q^{161}-7 q^{159}-389 q^{157}-415 q^{155}-89 q^{153}+291 q^{151}+419 q^{149}+199 q^{147}-177 q^{145}-426 q^{143}-345 q^{141}+44 q^{139}+437 q^{137}+497 q^{135}+130 q^{133}-414 q^{131}-656 q^{129}-324 q^{127}+342 q^{125}+771 q^{123}+538 q^{121}-213 q^{119}-828 q^{117}-719 q^{115}+35 q^{113}+775 q^{111}+850 q^{109}+160 q^{107}-656 q^{105}-871 q^{103}-332 q^{101}+455 q^{99}+800 q^{97}+446 q^{95}-249 q^{93}-656 q^{91}-466 q^{89}+70 q^{87}+461 q^{85}+418 q^{83}+59 q^{81}-290 q^{79}-326 q^{77}-96 q^{75}+139 q^{73}+214 q^{71}+107 q^{69}-55 q^{67}-129 q^{65}-76 q^{63}+13 q^{61}+62 q^{59}+49 q^{57}-27 q^{53}-23 q^{51}-3 q^{49}+15 q^{47}+11 q^{45}-q^{43}-6 q^{41}-q^{37}+4 q^{35}+5 q^{33}-2 q^{31}-2 q^{29}+2 q^{27}+2 q^{21}-q^{17}+q^{15} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{34}-2 q^{28}+q^{26}+q^{20}-q^{18}+2 q^{16}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
1,1	[math]\displaystyle{ q^{92}-2 q^{90}+6 q^{88}-12 q^{86}+23 q^{84}-40 q^{82}+60 q^{80}-84 q^{78}+110 q^{76}-132 q^{74}+142 q^{72}-136 q^{70}+114 q^{68}-76 q^{66}+18 q^{64}+52 q^{62}-119 q^{60}+184 q^{58}-236 q^{56}+268 q^{54}-280 q^{52}+260 q^{50}-224 q^{48}+162 q^{46}-97 q^{44}+24 q^{42}+38 q^{40}-92 q^{38}+123 q^{36}-134 q^{34}+136 q^{32}-120 q^{30}+103 q^{28}-72 q^{26}+56 q^{24}-34 q^{22}+23 q^{20}-10 q^{18}+6 q^{16}-2 q^{14}+q^{12} }[/math]
2,0	[math]\displaystyle{ q^{86}+2 q^{78}-4 q^{74}-q^{72}+4 q^{70}+2 q^{68}-4 q^{66}-q^{64}+5 q^{62}+q^{60}-7 q^{58}-2 q^{56}+2 q^{54}-3 q^{52}-3 q^{50}+q^{48}+q^{46}-3 q^{44}+q^{42}+3 q^{40}-4 q^{38}-q^{36}+7 q^{34}+2 q^{32}-5 q^{30}+3 q^{28}+7 q^{26}+q^{24}-4 q^{22}+2 q^{20}+4 q^{18}-q^{16}-q^{14}+q^{12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{74}-q^{72}+q^{70}+2 q^{68}-4 q^{66}+2 q^{64}+4 q^{62}-8 q^{60}+3 q^{58}+5 q^{56}-10 q^{54}+q^{52}+5 q^{50}-6 q^{48}-2 q^{46}+2 q^{44}-3 q^{40}-2 q^{38}+7 q^{36}-2 q^{34}-5 q^{32}+10 q^{30}-6 q^{26}+9 q^{24}+q^{22}-3 q^{20}+4 q^{18}+q^{16}-q^{14}+q^{12} }[/math]
1,0,0	[math]\displaystyle{ -q^{45}-q^{41}-2 q^{37}+q^{35}-q^{33}+q^{31}+q^{27}+2 q^{21}+2 q^{17}+2 q^{13}-q^{11}+q^9 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{96}-q^{92}+2 q^{90}+3 q^{88}-2 q^{86}-q^{84}+5 q^{82}+3 q^{80}-6 q^{78}-4 q^{76}+4 q^{74}-3 q^{72}-10 q^{70}+4 q^{66}-5 q^{64}-3 q^{62}+4 q^{60}-q^{58}-6 q^{56}+q^{54}+4 q^{52}-5 q^{50}-2 q^{48}+9 q^{46}+2 q^{44}-5 q^{42}+4 q^{40}+8 q^{38}-2 q^{34}+4 q^{32}+5 q^{30}-q^{28}+3 q^{24}+q^{22}-q^{20}+q^{18} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{56}-q^{52}-q^{50}-2 q^{46}+q^{44}-q^{42}+q^{38}+q^{34}+q^{30}+2 q^{26}+2 q^{22}+q^{20}+2 q^{16}-q^{14}+q^{12} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{74}+q^{72}-3 q^{70}+4 q^{68}-6 q^{66}+8 q^{64}-8 q^{62}+8 q^{60}-7 q^{58}+5 q^{56}-2 q^{54}-3 q^{52}+7 q^{50}-12 q^{48}+14 q^{46}-16 q^{44}+16 q^{42}-15 q^{40}+12 q^{38}-7 q^{36}+4 q^{34}+q^{32}-4 q^{30}+8 q^{28}-8 q^{26}+9 q^{24}-7 q^{22}+7 q^{20}-4 q^{18}+3 q^{16}-q^{14}+q^{12} }[/math]
1,0	[math]\displaystyle{ q^{120}-q^{116}-q^{114}+2 q^{112}+3 q^{110}-q^{108}-5 q^{106}-2 q^{104}+6 q^{102}+6 q^{100}-4 q^{98}-9 q^{96}-q^{94}+9 q^{92}+5 q^{90}-7 q^{88}-8 q^{86}+2 q^{84}+7 q^{82}-7 q^{78}-2 q^{76}+5 q^{74}+2 q^{72}-6 q^{70}-4 q^{68}+4 q^{66}+5 q^{64}-3 q^{62}-6 q^{60}+2 q^{58}+7 q^{56}+q^{54}-8 q^{52}-3 q^{50}+8 q^{48}+8 q^{46}-4 q^{44}-8 q^{42}+9 q^{38}+5 q^{36}-3 q^{34}-5 q^{32}+q^{30}+4 q^{28}+2 q^{26}-q^{24}-q^{22}+q^{18} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{102}-q^{100}+2 q^{98}-2 q^{96}+4 q^{94}-5 q^{92}+5 q^{90}-6 q^{88}+7 q^{86}-7 q^{84}+5 q^{82}-5 q^{80}+4 q^{78}-3 q^{76}-3 q^{74}+2 q^{72}-5 q^{70}+7 q^{68}-12 q^{66}+10 q^{64}-12 q^{62}+13 q^{60}-13 q^{58}+9 q^{56}-10 q^{54}+9 q^{52}-4 q^{50}+3 q^{48}-q^{46}+7 q^{42}-4 q^{40}+6 q^{38}-6 q^{36}+9 q^{34}-4 q^{32}+6 q^{30}-4 q^{28}+5 q^{26}-q^{24}+2 q^{22}-q^{20}+q^{18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{176}-q^{174}+3 q^{172}-4 q^{170}+3 q^{168}-2 q^{166}-2 q^{164}+10 q^{162}-15 q^{160}+18 q^{158}-15 q^{156}+5 q^{154}+9 q^{152}-26 q^{150}+36 q^{148}-37 q^{146}+23 q^{144}-2 q^{142}-24 q^{140}+37 q^{138}-38 q^{136}+27 q^{134}-6 q^{132}-16 q^{130}+24 q^{128}-23 q^{126}+5 q^{124}+16 q^{122}-29 q^{120}+31 q^{118}-16 q^{116}-7 q^{114}+34 q^{112}-51 q^{110}+54 q^{108}-40 q^{106}+9 q^{104}+23 q^{102}-48 q^{100}+58 q^{98}-47 q^{96}+23 q^{94}+4 q^{92}-26 q^{90}+32 q^{88}-25 q^{86}+4 q^{84}+16 q^{82}-23 q^{80}+20 q^{78}-2 q^{76}-18 q^{74}+35 q^{72}-36 q^{70}+29 q^{68}-12 q^{66}-11 q^{64}+29 q^{62}-34 q^{60}+33 q^{58}-18 q^{56}+6 q^{54}+6 q^{52}-14 q^{50}+16 q^{48}-13 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/math]

.

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 18"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 4 t^2-10 t+13-10 t^{-1} +4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 4 z^4+6 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 41, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +7 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^{10}-a^{10}+z^4 a^8+z^2 a^8+2 z^4 a^6+4 z^2 a^6+a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-5 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-3 z^5 a^{11}+z^8 a^{10}+z^6 a^{10}-2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+4 z^7 a^9-5 z^5 a^9+z^3 a^9+z^8 a^8+2 z^6 a^8-2 z^4 a^8+2 z^7 a^7+z^5 a^7-4 z^3 a^7+2 z a^7+3 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a246, ...}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {...}

Vassiliev invariants

V2 and V3:

(6, -15)

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 [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ -120 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 748 }[/math] [math]\displaystyle{ 108 }[/math] [math]\displaystyle{ -2880 }[/math] [math]\displaystyle{ -5200 }[/math] [math]\displaystyle{ -896 }[/math] [math]\displaystyle{ -664 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 7200 }[/math] [math]\displaystyle{ 17952 }[/math] [math]\displaystyle{ 2592 }[/math] [math]\displaystyle{ \frac{184471}{5} }[/math] [math]\displaystyle{ \frac{17828}{15} }[/math] [math]\displaystyle{ \frac{207524}{15} }[/math] [math]\displaystyle{ \frac{889}{3} }[/math] [math]\displaystyle{ \frac{8791}{5} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-4 is the signature of 9 18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                   1 1 -5                 2 1 -1 -7               3     3 -9             3 2     -1 -11           4 3       1 -13         3 3         0 -15       3 4           -1 -17     1 3             2 -19   1 3               -2 -21   1                 1 -23 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	[math]\displaystyle{ q^{-4} -2 q^{-5} + q^{-6} +6 q^{-7} -10 q^{-8} + q^{-9} +20 q^{-10} -25 q^{-11} -3 q^{-12} +39 q^{-13} -37 q^{-14} -10 q^{-15} +52 q^{-16} -39 q^{-17} -16 q^{-18} +51 q^{-19} -30 q^{-20} -19 q^{-21} +38 q^{-22} -15 q^{-23} -15 q^{-24} +20 q^{-25} -4 q^{-26} -8 q^{-27} +6 q^{-28} -2 q^{-30} + q^{-31} }[/math]
3	[math]\displaystyle{ q^{-6} -2 q^{-7} + q^{-8} +2 q^{-9} +2 q^{-10} -8 q^{-11} +13 q^{-13} +5 q^{-14} -27 q^{-15} -5 q^{-16} +37 q^{-17} +22 q^{-18} -65 q^{-19} -29 q^{-20} +78 q^{-21} +57 q^{-22} -104 q^{-23} -76 q^{-24} +114 q^{-25} +107 q^{-26} -128 q^{-27} -126 q^{-28} +127 q^{-29} +145 q^{-30} -124 q^{-31} -155 q^{-32} +111 q^{-33} +160 q^{-34} -95 q^{-35} -157 q^{-36} +74 q^{-37} +148 q^{-38} -50 q^{-39} -134 q^{-40} +26 q^{-41} +116 q^{-42} -7 q^{-43} -93 q^{-44} -7 q^{-45} +68 q^{-46} +18 q^{-47} -48 q^{-48} -17 q^{-49} +26 q^{-50} +17 q^{-51} -15 q^{-52} -10 q^{-53} +5 q^{-54} +7 q^{-55} -3 q^{-56} -2 q^{-57} +2 q^{-59} - q^{-60} }[/math]
4	[math]\displaystyle{ q^{-8} -2 q^{-9} + q^{-10} +2 q^{-11} -2 q^{-12} +4 q^{-13} -9 q^{-14} +3 q^{-15} +11 q^{-16} -7 q^{-17} +9 q^{-18} -31 q^{-19} +9 q^{-20} +39 q^{-21} -12 q^{-22} +10 q^{-23} -89 q^{-24} +15 q^{-25} +106 q^{-26} +10 q^{-27} +14 q^{-28} -221 q^{-29} -15 q^{-30} +218 q^{-31} +103 q^{-32} +53 q^{-33} -423 q^{-34} -123 q^{-35} +321 q^{-36} +266 q^{-37} +168 q^{-38} -626 q^{-39} -300 q^{-40} +355 q^{-41} +433 q^{-42} +333 q^{-43} -748 q^{-44} -465 q^{-45} +310 q^{-46} +527 q^{-47} +486 q^{-48} -762 q^{-49} -557 q^{-50} +218 q^{-51} +529 q^{-52} +582 q^{-53} -679 q^{-54} -567 q^{-55} +97 q^{-56} +456 q^{-57} +621 q^{-58} -520 q^{-59} -512 q^{-60} -36 q^{-61} +323 q^{-62} +599 q^{-63} -308 q^{-64} -397 q^{-65} -153 q^{-66} +155 q^{-67} +506 q^{-68} -105 q^{-69} -239 q^{-70} -195 q^{-71} +4 q^{-72} +343 q^{-73} +19 q^{-74} -83 q^{-75} -151 q^{-76} -73 q^{-77} +171 q^{-78} +42 q^{-79} +7 q^{-80} -70 q^{-81} -67 q^{-82} +58 q^{-83} +17 q^{-84} +23 q^{-85} -17 q^{-86} -31 q^{-87} +15 q^{-88} +10 q^{-90} - q^{-91} -9 q^{-92} +4 q^{-93} - q^{-94} +2 q^{-95} -2 q^{-97} + q^{-98} }[/math]
5	Not Available
6	Not Available
7	Not Available

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, 18]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[9, 18, 10, 1], X[17, 6, 18, 7], X[7, 16, 8, 17], X[15, 8, 16, 9], X[13, 10, 14, 11], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[9, 18]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4]
In[4]:=	DTCode[Knot[9, 18]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 2, 10, 8, 6]
In[5]:=	br = BR[Knot[9, 18]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, -2, -2, -2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[9, 18]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 18]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 18]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 18]][t]
Out[10]=	4 10 2 13 + -- - -- - 10 t + 4 t 2 t t
In[11]:=	Conway[Knot[9, 18]][z]
Out[11]=	2 4 1 + 6 z + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 18], Knot[11, Alternating, 246]}
In[13]:=	{KnotDet[Knot[9, 18]], KnotSignature[Knot[9, 18]]}
Out[13]=	{41, -4}
In[14]:=	Jones[Knot[9, 18]][q]
Out[14]=	-11 2 4 6 7 7 6 5 2 -2 -q + --- - -- + -- - -- + -- - -- + -- - -- + q 10 9 8 7 6 5 4 3 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 18]}
In[16]:=	A2Invariant[Knot[9, 18]][q]
Out[16]=	-34 2 -26 -20 -18 2 -12 2 -8 -6 -q - --- + q + q - q + --- + q + --- - q + q 28 16 10 q q q
In[17]:=	HOMFLYPT[Knot[9, 18]][a, z]
Out[17]=	4 6 10 4 2 6 2 8 2 10 2 4 4 6 4 a + a - a + 2 a z + 4 a z + a z - a z + a z + 2 a z + 8 4 a z
In[18]:=	Kauffman[Knot[9, 18]][a, z]
Out[18]=	4 6 10 7 13 4 2 6 2 10 2 a - a + a + 2 a z + 2 a z - 2 a z + 3 a z - 2 a z + 12 2 5 3 7 3 9 3 13 3 4 4 6 4 3 a z - 2 a z - 4 a z + a z - 3 a z + a z - 4 a z - 8 4 10 4 12 4 5 5 7 5 9 5 2 a z - 2 a z - 5 a z + 2 a z + a z - 5 a z - 11 5 13 5 6 6 8 6 10 6 12 6 7 7 3 a z + a z + 3 a z + 2 a z + a z + 2 a z + 2 a z + 9 7 11 7 8 8 10 8 4 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 18]], Vassiliev[3][Knot[9, 18]]}
Out[19]=	{6, -15}
In[20]:=	Kh[Knot[9, 18]][q, t]
Out[20]=	-5 -3 1 1 1 3 1 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 21 8 19 8 19 7 17 7 17 6 q t q t q t q t q t q t 3 4 3 3 4 3 3 2 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 15 6 15 5 13 5 13 4 11 4 11 3 9 3 9 2 q t q t q t q t q t q t q t q t 3 2 ----- + ---- 7 2 5 q t q t
In[21]:=	ColouredJones[Knot[9, 18], 2][q]
Out[21]=	-31 2 6 8 4 20 15 15 38 19 30 q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + 30 28 27 26 25 24 23 22 21 20 q q q q q q q q q q 51 16 39 52 10 37 39 3 25 20 -9 --- - --- - --- + --- - --- - --- + --- - --- - --- + --- + q - 19 18 17 16 15 14 13 12 11 10 q q q q q q q q q q 10 6 -6 2 -4 -- + -- + q - -- + q 8 7 5 q q q

See/edit the Rolfsen_Splice_Template.