9 23

9_22

9_24

Visit 9 23's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 23's page at Knotilus! Visit 9 23's page at the original Knot Atlas! 9 23 Quick Notes

Symmetrical decorative knot	With crossings on 3x3 grid	Depiction with two axes of symmetry	Mongolian ornament (two crossings are unnecessary)
Mongolian ornament, sum of two 9.23	Logo of the ICMC-USP, Brazil	Other depicture with central symmetry by Alain Esculier

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,12,6,13 X7,16,8,17 X13,18,14,1 X17,14,18,15 X15,6,16,7 X11,8,12,9 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5
Dowker-Thistlethwaite code	4 10 12 16 2 8 18 6 14
Conway Notation	[22122]

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{ \{4,7\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 10.6113 A-Polynomial See Data:9 23/A-polynomial

[edit Notes for 9 23'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 23's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t^2-11 t+15-11 t^{-1} +4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 45, -4 }
Jones polynomial	[math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +8 q^{-6} -8 q^{-7} +6 q^{-8} -5 q^{-9} +3 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}+z^4 a^8-2 a^8+2 z^4 a^6+4 z^2 a^6+2 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}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-7 z^4 a^{12}+3 z^2 a^{12}+3 z^7 a^{11}-5 z^5 a^{11}+z a^{11}+z^8 a^{10}+4 z^6 a^{10}-10 z^4 a^{10}+3 z^2 a^{10}+5 z^7 a^9-6 z^5 a^9-2 z^3 a^9+4 z a^9+z^8 a^8+4 z^6 a^8-8 z^4 a^8+6 z^2 a^8-2 a^8+2 z^7 a^7+2 z^5 a^7-6 z^3 a^7+4 z a^7+3 z^6 a^6-4 z^4 a^6+4 z^2 a^6-2 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}+q^{32}+q^{30}-2 q^{28}-2 q^{24}-q^{22}+q^{20}+3 q^{16}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{176}-2 q^{174}+5 q^{172}-8 q^{170}+7 q^{168}-3 q^{166}-6 q^{164}+20 q^{162}-27 q^{160}+30 q^{158}-24 q^{156}+q^{154}+23 q^{152}-47 q^{150}+55 q^{148}-43 q^{146}+18 q^{144}+14 q^{142}-38 q^{140}+48 q^{138}-38 q^{136}+13 q^{134}+12 q^{132}-30 q^{130}+30 q^{128}-10 q^{126}-16 q^{124}+40 q^{122}-43 q^{120}+36 q^{118}-13 q^{116}-28 q^{114}+56 q^{112}-73 q^{110}+66 q^{108}-39 q^{106}-6 q^{104}+43 q^{102}-64 q^{100}+61 q^{98}-44 q^{96}+7 q^{94}+20 q^{92}-36 q^{90}+30 q^{88}-7 q^{86}-16 q^{84}+35 q^{82}-29 q^{80}+11 q^{78}+13 q^{76}-34 q^{74}+47 q^{72}-41 q^{70}+28 q^{68}-2 q^{66}-19 q^{64}+35 q^{62}-35 q^{60}+31 q^{58}-17 q^{56}+4 q^{54}+7 q^{52}-15 q^{50}+16 q^{48}-12 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_23.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{23}+2 q^{21}-2 q^{19}+q^{17}-2 q^{15}+2 q^{11}-q^9+3 q^7-q^5+q^3 }[/math]
2	[math]\displaystyle{ q^{64}-2 q^{62}-2 q^{60}+7 q^{58}-q^{56}-9 q^{54}+9 q^{52}+3 q^{50}-13 q^{48}+6 q^{46}+7 q^{44}-10 q^{42}+2 q^{40}+7 q^{38}-3 q^{36}-6 q^{34}+q^{32}+7 q^{30}-10 q^{28}-4 q^{26}+14 q^{24}-6 q^{22}-6 q^{20}+11 q^{18}-2 q^{16}-3 q^{14}+5 q^{12}-q^8+q^6 }[/math]
3	[math]\displaystyle{ -q^{123}+2 q^{121}+2 q^{119}-3 q^{117}-7 q^{115}+q^{113}+16 q^{111}+4 q^{109}-21 q^{107}-15 q^{105}+22 q^{103}+29 q^{101}-20 q^{99}-41 q^{97}+10 q^{95}+49 q^{93}+3 q^{91}-50 q^{89}-16 q^{87}+50 q^{85}+24 q^{83}-41 q^{81}-30 q^{79}+32 q^{77}+32 q^{75}-23 q^{73}-32 q^{71}+8 q^{69}+33 q^{67}+5 q^{65}-24 q^{63}-25 q^{61}+21 q^{59}+38 q^{57}-10 q^{55}-51 q^{53}-3 q^{51}+53 q^{49}+14 q^{47}-52 q^{45}-23 q^{43}+40 q^{41}+24 q^{39}-28 q^{37}-23 q^{35}+20 q^{33}+16 q^{31}-9 q^{29}-10 q^{27}+8 q^{25}+7 q^{23}-3 q^{21}-3 q^{19}+3 q^{17}+2 q^{15}-q^{11}+q^9 }[/math]
4	[math]\displaystyle{ q^{200}-2 q^{198}-2 q^{196}+3 q^{194}+3 q^{192}+7 q^{190}-8 q^{188}-16 q^{186}-4 q^{184}+8 q^{182}+42 q^{180}+8 q^{178}-38 q^{176}-47 q^{174}-26 q^{172}+80 q^{170}+74 q^{168}-2 q^{166}-91 q^{164}-129 q^{162}+48 q^{160}+141 q^{158}+111 q^{156}-53 q^{154}-222 q^{152}-72 q^{150}+115 q^{148}+222 q^{146}+68 q^{144}-223 q^{142}-187 q^{140}+12 q^{138}+241 q^{136}+171 q^{134}-143 q^{132}-218 q^{130}-80 q^{128}+186 q^{126}+200 q^{124}-53 q^{122}-179 q^{120}-116 q^{118}+108 q^{116}+175 q^{114}+23 q^{112}-124 q^{110}-133 q^{108}+16 q^{106}+128 q^{104}+114 q^{102}-40 q^{100}-145 q^{98}-109 q^{96}+55 q^{94}+212 q^{92}+80 q^{90}-120 q^{88}-226 q^{86}-64 q^{84}+234 q^{82}+193 q^{80}-21 q^{78}-254 q^{76}-175 q^{74}+153 q^{72}+208 q^{70}+81 q^{68}-167 q^{66}-188 q^{64}+43 q^{62}+122 q^{60}+103 q^{58}-58 q^{56}-115 q^{54}-3 q^{52}+36 q^{50}+61 q^{48}-8 q^{46}-44 q^{44}+4 q^{40}+22 q^{38}-2 q^{36}-13 q^{34}+6 q^{32}+7 q^{28}-q^{26}-4 q^{24}+3 q^{22}+2 q^{18}-q^{14}+q^{12} }[/math]
5	[math]\displaystyle{ -q^{295}+2 q^{293}+2 q^{291}-3 q^{289}-3 q^{287}-3 q^{285}+8 q^{281}+16 q^{279}+4 q^{277}-17 q^{275}-29 q^{273}-26 q^{271}+8 q^{269}+54 q^{267}+74 q^{265}+19 q^{263}-70 q^{261}-124 q^{259}-94 q^{257}+38 q^{255}+183 q^{253}+209 q^{251}+46 q^{249}-195 q^{247}-327 q^{245}-217 q^{243}+118 q^{241}+429 q^{239}+433 q^{237}+53 q^{235}-435 q^{233}-637 q^{231}-334 q^{229}+320 q^{227}+792 q^{225}+648 q^{223}-82 q^{221}-822 q^{219}-937 q^{217}-253 q^{215}+725 q^{213}+1141 q^{211}+595 q^{209}-506 q^{207}-1226 q^{205}-894 q^{203}+234 q^{201}+1175 q^{199}+1097 q^{197}+55 q^{195}-1039 q^{193}-1188 q^{191}-273 q^{189}+841 q^{187}+1168 q^{185}+438 q^{183}-642 q^{181}-1082 q^{179}-522 q^{177}+464 q^{175}+946 q^{173}+545 q^{171}-302 q^{169}-814 q^{167}-563 q^{165}+174 q^{163}+686 q^{161}+565 q^{159}-13 q^{157}-553 q^{155}-629 q^{153}-153 q^{151}+432 q^{149}+669 q^{147}+394 q^{145}-249 q^{143}-752 q^{141}-648 q^{139}+33 q^{137}+765 q^{135}+919 q^{133}+263 q^{131}-717 q^{129}-1145 q^{127}-588 q^{125}+552 q^{123}+1285 q^{121}+894 q^{119}-303 q^{117}-1273 q^{115}-1131 q^{113}-4 q^{111}+1127 q^{109}+1242 q^{107}+280 q^{105}-859 q^{103}-1192 q^{101}-493 q^{99}+546 q^{97}+1019 q^{95}+593 q^{93}-269 q^{91}-766 q^{89}-564 q^{87}+43 q^{85}+502 q^{83}+475 q^{81}+66 q^{79}-295 q^{77}-329 q^{75}-112 q^{73}+136 q^{71}+212 q^{69}+96 q^{67}-57 q^{65}-112 q^{63}-66 q^{61}+16 q^{59}+55 q^{57}+34 q^{55}-2 q^{53}-19 q^{51}-16 q^{49}+3 q^{47}+10 q^{45}+3 q^{43}-2 q^{41}+2 q^{39}-2 q^{37}+2 q^{35}+5 q^{33}-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}+q^{32}+q^{30}-2 q^{28}-2 q^{24}-q^{22}+q^{20}+3 q^{16}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
1,1	[math]\displaystyle{ q^{92}-4 q^{90}+12 q^{88}-28 q^{86}+50 q^{84}-80 q^{82}+118 q^{80}-154 q^{78}+184 q^{76}-194 q^{74}+186 q^{72}-156 q^{70}+93 q^{68}-18 q^{66}-74 q^{64}+166 q^{62}-244 q^{60}+316 q^{58}-352 q^{56}+364 q^{54}-340 q^{52}+284 q^{50}-212 q^{48}+120 q^{46}-36 q^{44}-54 q^{42}+108 q^{40}-152 q^{38}+165 q^{36}-166 q^{34}+154 q^{32}-124 q^{30}+105 q^{28}-72 q^{26}+56 q^{24}-32 q^{22}+23 q^{20}-10 q^{18}+6 q^{16}-2 q^{14}+q^{12} }[/math]
2,0	[math]\displaystyle{ q^{86}-q^{84}-2 q^{82}+4 q^{78}+3 q^{76}-5 q^{74}-2 q^{72}+4 q^{70}+2 q^{68}-5 q^{66}-3 q^{64}+7 q^{62}+2 q^{60}-4 q^{58}+5 q^{54}-2 q^{52}-4 q^{50}-q^{48}-4 q^{46}-6 q^{44}-q^{42}+3 q^{40}-6 q^{38}-q^{36}+10 q^{34}+4 q^{32}-5 q^{30}+2 q^{28}+8 q^{26}+2 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}-2 q^{72}+q^{70}+3 q^{68}-6 q^{66}+4 q^{64}+4 q^{62}-10 q^{60}+5 q^{58}+5 q^{56}-9 q^{54}+3 q^{52}+8 q^{50}-5 q^{48}-2 q^{46}-2 q^{42}-7 q^{40}-6 q^{38}+8 q^{36}-4 q^{34}-4 q^{32}+13 q^{30}-5 q^{26}+10 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^{43}+q^{39}-2 q^{37}-3 q^{33}-q^{31}-q^{29}+q^{27}+q^{25}+q^{23}+3 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^{94}-2 q^{92}+3 q^{90}+3 q^{88}-4 q^{86}-2 q^{84}+6 q^{82}+q^{80}-8 q^{78}-q^{76}+9 q^{74}-6 q^{70}+7 q^{68}+6 q^{66}-6 q^{64}-3 q^{62}+2 q^{60}-8 q^{58}-11 q^{56}-2 q^{54}-2 q^{52}-10 q^{50}-2 q^{48}+11 q^{46}+2 q^{44}-3 q^{42}+8 q^{40}+10 q^{38}-q^{34}+5 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^{54}+q^{48}-2 q^{46}-3 q^{42}-2 q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+2 q^{30}+q^{28}+3 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}+2 q^{72}-5 q^{70}+7 q^{68}-8 q^{66}+10 q^{64}-10 q^{62}+10 q^{60}-7 q^{58}+3 q^{56}+q^{54}-7 q^{52}+10 q^{50}-17 q^{48}+18 q^{46}-20 q^{44}+18 q^{42}-15 q^{40}+12 q^{38}-6 q^{36}+2 q^{34}+4 q^{32}-7 q^{30}+10 q^{28}-9 q^{26}+10 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}-2 q^{116}-2 q^{114}+3 q^{112}+5 q^{110}-2 q^{108}-7 q^{106}-2 q^{104}+9 q^{102}+7 q^{100}-7 q^{98}-11 q^{96}+q^{94}+11 q^{92}+4 q^{90}-8 q^{88}-7 q^{86}+5 q^{84}+8 q^{82}-7 q^{78}+6 q^{74}+q^{72}-9 q^{70}-5 q^{68}+5 q^{66}+3 q^{64}-7 q^{62}-8 q^{60}+4 q^{58}+8 q^{56}-q^{54}-10 q^{52}-q^{50}+11 q^{48}+8 q^{46}-5 q^{44}-8 q^{42}+q^{40}+10 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}-2 q^{100}+3 q^{98}-4 q^{96}+6 q^{94}-7 q^{92}+7 q^{90}-8 q^{88}+9 q^{86}-8 q^{84}+5 q^{82}-4 q^{80}+3 q^{78}-4 q^{74}+7 q^{72}-6 q^{70}+12 q^{68}-14 q^{66}+13 q^{64}-16 q^{62}+13 q^{60}-18 q^{58}+6 q^{56}-13 q^{54}+6 q^{52}-4 q^{50}+q^{48}+2 q^{46}-q^{44}+11 q^{42}-4 q^{40}+8 q^{38}-6 q^{36}+10 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}-2 q^{174}+5 q^{172}-8 q^{170}+7 q^{168}-3 q^{166}-6 q^{164}+20 q^{162}-27 q^{160}+30 q^{158}-24 q^{156}+q^{154}+23 q^{152}-47 q^{150}+55 q^{148}-43 q^{146}+18 q^{144}+14 q^{142}-38 q^{140}+48 q^{138}-38 q^{136}+13 q^{134}+12 q^{132}-30 q^{130}+30 q^{128}-10 q^{126}-16 q^{124}+40 q^{122}-43 q^{120}+36 q^{118}-13 q^{116}-28 q^{114}+56 q^{112}-73 q^{110}+66 q^{108}-39 q^{106}-6 q^{104}+43 q^{102}-64 q^{100}+61 q^{98}-44 q^{96}+7 q^{94}+20 q^{92}-36 q^{90}+30 q^{88}-7 q^{86}-16 q^{84}+35 q^{82}-29 q^{80}+11 q^{78}+13 q^{76}-34 q^{74}+47 q^{72}-41 q^{70}+28 q^{68}-2 q^{66}-19 q^{64}+35 q^{62}-35 q^{60}+31 q^{58}-17 q^{56}+4 q^{54}+7 q^{52}-15 q^{50}+16 q^{48}-12 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 23"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 4 t^2-11 t+15-11 t^{-1} +4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 4 z^4+5 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]=

{ 45, -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} +8 q^{-6} -8 q^{-7} +6 q^{-8} -5 q^{-9} +3 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}+z^4 a^8-2 a^8+2 z^4 a^6+4 z^2 a^6+2 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}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-7 z^4 a^{12}+3 z^2 a^{12}+3 z^7 a^{11}-5 z^5 a^{11}+z a^{11}+z^8 a^{10}+4 z^6 a^{10}-10 z^4 a^{10}+3 z^2 a^{10}+5 z^7 a^9-6 z^5 a^9-2 z^3 a^9+4 z a^9+z^8 a^8+4 z^6 a^8-8 z^4 a^8+6 z^2 a^8-2 a^8+2 z^7 a^7+2 z^5 a^7-6 z^3 a^7+4 z a^7+3 z^6 a^6-4 z^4 a^6+4 z^2 a^6-2 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: {...}

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

Vassiliev invariants

V2 and V3:

(5, -11)

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{ 20 }[/math] [math]\displaystyle{ -88 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1462}{3} }[/math] [math]\displaystyle{ \frac{194}{3} }[/math] [math]\displaystyle{ -1760 }[/math] [math]\displaystyle{ -\frac{9040}{3} }[/math] [math]\displaystyle{ -\frac{1504}{3} }[/math] [math]\displaystyle{ -344 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 3872 }[/math] [math]\displaystyle{ \frac{29240}{3} }[/math] [math]\displaystyle{ \frac{3880}{3} }[/math] [math]\displaystyle{ \frac{115087}{6} }[/math] [math]\displaystyle{ \frac{2986}{3} }[/math] [math]\displaystyle{ \frac{58214}{9} }[/math] [math]\displaystyle{ \frac{2389}{18} }[/math] [math]\displaystyle{ \frac{4687}{6} }[/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 23. 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           5 3       2 -13         3 3         0 -15       3 5           -2 -17     2 3             1 -19   1 3               -2 -21   2                 2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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} +2 q^{-9} +19 q^{-10} -27 q^{-11} +2 q^{-12} +39 q^{-13} -45 q^{-14} -4 q^{-15} +56 q^{-16} -51 q^{-17} -11 q^{-18} +59 q^{-19} -41 q^{-20} -16 q^{-21} +47 q^{-22} -24 q^{-23} -17 q^{-24} +28 q^{-25} -8 q^{-26} -11 q^{-27} +10 q^{-28} -3 q^{-30} + q^{-31} }[/math]
3	[math]\displaystyle{ q^{-6} -2 q^{-7} + q^{-8} +2 q^{-9} +2 q^{-10} -8 q^{-11} + q^{-12} +12 q^{-13} +3 q^{-14} -26 q^{-15} +2 q^{-16} +37 q^{-17} +7 q^{-18} -69 q^{-19} -3 q^{-20} +89 q^{-21} +23 q^{-22} -132 q^{-23} -32 q^{-24} +155 q^{-25} +62 q^{-26} -188 q^{-27} -80 q^{-28} +196 q^{-29} +110 q^{-30} -205 q^{-31} -126 q^{-32} +197 q^{-33} +139 q^{-34} -177 q^{-35} -151 q^{-36} +157 q^{-37} +148 q^{-38} -122 q^{-39} -151 q^{-40} +95 q^{-41} +137 q^{-42} -57 q^{-43} -125 q^{-44} +29 q^{-45} +103 q^{-46} -4 q^{-47} -79 q^{-48} -10 q^{-49} +52 q^{-50} +17 q^{-51} -30 q^{-52} -17 q^{-53} +15 q^{-54} +11 q^{-55} -5 q^{-56} -5 q^{-57} +3 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} +4 q^{-15} +10 q^{-16} -9 q^{-17} +10 q^{-18} -28 q^{-19} +15 q^{-20} +34 q^{-21} -27 q^{-22} +6 q^{-23} -72 q^{-24} +51 q^{-25} +103 q^{-26} -52 q^{-27} -33 q^{-28} -184 q^{-29} +108 q^{-30} +264 q^{-31} -33 q^{-32} -112 q^{-33} -415 q^{-34} +129 q^{-35} +512 q^{-36} +94 q^{-37} -167 q^{-38} -743 q^{-39} +50 q^{-40} +745 q^{-41} +308 q^{-42} -126 q^{-43} -1041 q^{-44} -112 q^{-45} +851 q^{-46} +508 q^{-47} +6 q^{-48} -1198 q^{-49} -276 q^{-50} +815 q^{-51} +613 q^{-52} +160 q^{-53} -1184 q^{-54} -388 q^{-55} +666 q^{-56} +622 q^{-57} +307 q^{-58} -1032 q^{-59} -455 q^{-60} +442 q^{-61} +559 q^{-62} +433 q^{-63} -779 q^{-64} -469 q^{-65} +176 q^{-66} +421 q^{-67} +508 q^{-68} -465 q^{-69} -399 q^{-70} -53 q^{-71} +222 q^{-72} +472 q^{-73} -174 q^{-74} -245 q^{-75} -160 q^{-76} +35 q^{-77} +322 q^{-78} -5 q^{-79} -81 q^{-80} -130 q^{-81} -58 q^{-82} +145 q^{-83} +33 q^{-84} +8 q^{-85} -54 q^{-86} -52 q^{-87} +39 q^{-88} +12 q^{-89} +17 q^{-90} -8 q^{-91} -18 q^{-92} +5 q^{-93} +5 q^{-95} -3 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, 23]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 12, 6, 13], X[7, 16, 8, 17], X[13, 18, 14, 1], X[17, 14, 18, 15], X[15, 6, 16, 7], X[11, 8, 12, 9], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[9, 23]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5]
In[4]:=	DTCode[Knot[9, 23]]
Out[4]=	DTCode[4, 10, 12, 16, 2, 8, 18, 6, 14]
In[5]:=	br = BR[Knot[9, 23]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, -2, -2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[9, 23]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 23]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 23]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 23]][t]
Out[10]=	4 11 2 15 + -- - -- - 11 t + 4 t 2 t t
In[11]:=	Conway[Knot[9, 23]][z]
Out[11]=	2 4 1 + 5 z + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 23]}
In[13]:=	{KnotDet[Knot[9, 23]], KnotSignature[Knot[9, 23]]}
Out[13]=	{45, -4}
In[14]:=	Jones[Knot[9, 23]][q]
Out[14]=	-11 3 5 6 8 8 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, 23]}
In[16]:=	A2Invariant[Knot[9, 23]][q]
Out[16]=	-34 -32 -30 2 2 -22 -20 3 -12 2 -q + q + q - --- - --- - q + q + --- + q + --- - 28 24 16 10 q q q q -8 -6 q + q
In[17]:=	HOMFLYPT[Knot[9, 23]][a, z]
Out[17]=	4 6 8 4 2 6 2 10 2 4 4 6 4 8 4 a + 2 a - 2 a + 2 a z + 4 a z - a z + a z + 2 a z + a z
In[18]:=	Kauffman[Knot[9, 23]][a, z]
Out[18]=	4 6 8 7 9 11 13 4 2 a - 2 a - 2 a + 4 a z + 4 a z + a z + a z - 2 a z + 6 2 8 2 10 2 12 2 5 3 7 3 4 a z + 6 a z + 3 a z + 3 a z - 2 a z - 6 a z - 9 3 13 3 4 4 6 4 8 4 10 4 2 a z - 2 a z + a z - 4 a z - 8 a z - 10 a z - 12 4 5 5 7 5 9 5 11 5 13 5 7 a z + 2 a z + 2 a z - 6 a z - 5 a z + a z + 6 6 8 6 10 6 12 6 7 7 9 7 3 a z + 4 a z + 4 a z + 3 a z + 2 a z + 5 a z + 11 7 8 8 10 8 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 23]], Vassiliev[3][Knot[9, 23]]}
Out[19]=	{5, -11}
In[20]:=	Kh[Knot[9, 23]][q, t]
Out[20]=	-5 -3 1 2 1 3 2 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 5 3 3 5 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, 23], 2][q]
Out[21]=	-31 3 10 11 8 28 17 24 47 16 41 q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + 30 28 27 26 25 24 23 22 21 20 q q q q q q q q q q 59 11 51 56 4 45 39 2 27 19 2 10 --- - --- - --- + --- - --- - --- + --- + --- - --- + --- + -- - -- + 19 18 17 16 15 14 13 12 11 10 9 8 q q q q q q q q q q q q 6 -6 2 -4 -- + q - -- + q 7 5 q q

See/edit the Rolfsen_Splice_Template.