9 9

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9_8

9_10

Contents

Image:9 9.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 9's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9_9's page at Knotilus!

Visit 9 9's page at the original Knot Atlas!

[edit 9 9 Quick Notes]
[edit 9 9 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1627 X3,12,4,13 X7,16,8,17 X9,18,10,1 X17,8,18,9 X15,10,16,11 X5,14,6,15 X11,2,12,3 X13,4,14,5
Gauss code -1, 8, -2, 9, -7, 1, -3, 5, -4, 6, -8, 2, -9, 7, -6, 3, -5, 4
Dowker-Thistlethwaite code 6 12 14 16 18 2 4 10 8
Conway Notation [423]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
Image:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gif

Length is 10, width is 3,

Braid index is 3

Image:9 9_ML.gif Image:9 9_AP.gif
[{6, 1}, {11, 2}, {1, 3}, {2, 5}, {3, 7}, {4, 6}, {5, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 4}]

[edit Notes on presentations of 9 9]


Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["9 9"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1627 X3,12,4,13 X7,16,8,17 X9,18,10,1 X17,8,18,9 X15,10,16,11 X5,14,6,15 X11,2,12,3 X13,4,14,5
In[5]:= GaussCode[K]
Out[5]= -1, 8, -2, 9, -7, 1, -3, 5, -4, 6, -8, 2, -9, 7, -6, 3, -5, 4
In[6]:= DTCode[K]
Out[6]= 6 12 14 16 18 2 4 10 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [423]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= BR(3,{−1,−1,−1,−1,−1,−2,1,−2,−2,−2})
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 3, 10, 3 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
Image:9 9_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{6, 1}, {11, 2}, {1, 3}, {2, 5}, {3, 7}, {4, 6}, {5, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 4}]
In[14]:= Draw[ap]
Image:9 9_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index {4,6}
Nakanishi index 1
Maximal Thurston-Bennequin number [-16][5]
Hyperbolic Volume 8.01682
A-Polynomial See Data:9 9/A-polynomial

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

[edit] Four dimensional invariants

Smooth 4 genus 3
Topological 4 genus 3
Concordance genus 3
Rasmussen s-Invariant -6

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

[edit] Polynomial invariants

Alexander polynomial 2t3−4t2 + 6t−7 + 6t−1−4t−2 + 2t−3
Conway polynomial 2z6 + 8z4 + 8z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 31, -6 }
Jones polynomial q−3−q−4 + 3q−5−4q−6 + 5q−7−5q−8 + 5q−9−4q−10 + 2q−11−q−12
HOMFLY-PT polynomial (db, data sources) −z4a10−3z2a10−2a10 + z6a8 + 4z4a8 + 4z2a8 + a8 + z6a6 + 5z4a6 + 7z2a6 + 2a6
Kauffman polynomial (db, data sources) z3a15−za15 + 2z4a14−z2a14 + 3z5a13−3z3a13 + 2za13 + 3z6a12−4z4a12 + 3z2a12 + 2z7a11−2z5a11 + z8a10−z6a10 + 2z4a10−6z2a10 + 2a10 + 3z7a9−8z5a9 + 5z3a9−2za9 + z8a8−3z6a8 + 3z4a8−3z2a8 + a8 + z7a7−3z5a7 + z3a7 + za7 + z6a6−5z4a6 + 7z2a6−2a6
The A2 invariant −q36−q32−q30−q26 + 2q24 + q20 + q18 + 2q14 + q10
The G2 invariant q196−q194 + 2q192−2q190 + q188−2q184 + 5q182−6q180 + 6q178−6q176 + 2q174 + 3q172−7q170 + 12q168−11q166 + 9q164−5q162−2q160 + 6q158−11q156 + 11q154−7q152 + 3q148−7q146 + 5q144−q142−8q140 + 9q138−12q136 + 5q134 + 5q132−15q130 + 20q128−18q126 + 10q124 + q122−12q120 + 18q118−18q116 + 14q114−4q112−4q110 + 11q108−10q106 + 7q104−2q102−5q100 + 8q98−8q96 + 2q94 + 6q92−11q90 + 16q88−11q86 + q84 + 7q82−11q80 + 16q78−12q76 + 6q74 + 2q72−5q70 + 10q68−7q66 + 5q64 + 2q58−2q56 + 2q54 + q50
Further Quantum Invariants
Further quantum knot invariants for 9_9.

A1 Invariants.

Weight Invariant
1 −q25 + q23−2q21 + q19 + q13−q11 + 2q9 + q5
2 q68−q66 + 3q62−3q60−q58 + 5q56−5q54−2q52 + 5q50−2q48−2q46 + q44 + q42−3q40−2q38 + 4q36−4q32 + 4q30 + 2q28−5q26 + 2q24 + 3q22−3q20 + q18 + 3q16 + q10
3 −q129 + q127−q123−q121 + 2q119 + 2q117−3q115−q113 + 5q111−7q107−q105 + 11q103 + 2q101−13q99−2q97 + 12q95 + 6q93−10q91−6q89 + 5q87 + 8q85−q83−8q81−5q79 + 6q77 + 8q75−6q73−11q71 + 4q69 + 10q67−2q65−12q63−q61 + 11q59 + 3q57−11q55−6q53 + 9q51 + 10q49−6q47−10q45 + 3q43 + 10q41 + q39−9q37−3q35 + 5q33 + 5q31−2q29−2q27 + q25 + 3q23 + q21 + q15
4 q208−q206 + q202−q200 + 2q198−3q196−q194 + 2q192−2q190 + 5q188−4q186 + q184 + 4q182−9q180 + q178−3q176 + 13q174 + 12q172−19q170−13q168−10q166 + 26q164 + 32q162−18q160−31q158−28q156 + 25q154 + 46q152 + 2q150−24q148−37q146 + 2q144 + 34q142 + 21q140 + q138−26q136−20q134 + 2q132 + 21q130 + 23q128−5q126−26q124−19q122 + 15q120 + 29q118 + 8q116−26q114−29q112 + 14q110 + 29q108 + 13q106−27q104−33q102 + 11q100 + 27q98 + 25q96−19q94−39q92−5q90 + 18q88 + 37q86 + q84−33q82−22q80−6q78 + 35q76 + 24q74−10q72−23q70−26q68 + 14q66 + 26q64 + 13q62−5q60−26q58−7q56 + 8q54 + 14q52 + 9q50−10q48−8q46−4q44 + 4q42 + 8q40−2q34 + 3q30 + q28 + q26 + q20
5 −q305 + q303−q299 + q297−q293 + 2q291 + 2q289−2q287−q285−q283−3q281 + 2q279 + 4q277 + 4q275 + q273−3q271−8q269−12q267 + 16q263 + 24q261 + 9q259−22q257−43q255−27q253 + 27q251 + 72q249 + 52q247−26q245−94q243−86q241 + 6q239 + 113q237 + 124q235 + 21q233−114q231−154q229−57q227 + 88q225 + 165q223 + 98q221−56q219−151q217−117q215 + 2q213 + 112q211 + 125q209 + 39q207−63q205−102q203−68q201 + 7q199 + 75q197 + 80q195 + 32q193−36q191−78q189−60q187 + 8q185 + 73q183 + 75q181 + 9q179−67q177−78q175−13q173 + 67q171 + 82q169 + 14q167−73q165−88q163−11q161 + 82q159 + 95q157 + 19q155−88q153−112q151−28q149 + 84q147 + 124q145 + 49q143−71q141−129q139−77q137 + 41q135 + 124q133 + 100q131−6q129−102q127−115q125−38q123 + 67q121 + 115q119 + 70q117−22q115−92q113−94q111−27q109 + 57q107 + 93q105 + 59q103−14q101−71q99−78q97−27q95 + 40q93 + 70q91 + 47q89−q87−46q85−55q83−21q81 + 21q79 + 41q77 + 33q75 + 4q73−24q71−28q69−14q67 + 5q65 + 18q63 + 15q61 + 2q59−6q57−10q55−6q53 + 2q51 + 6q49 + 3q47 + 3q45−2q41 + 2q37 + q35 + q33 + q31 + q25
6 q420−q418 + q414−q412−q408 + 2q406−3q404−2q402 + 5q400 + q396 + 3q392−8q390−8q388 + 6q386 + q384 + 6q382 + 9q380 + 14q378−14q376−24q374−8q372−13q370 + 13q368 + 37q366 + 50q364−6q362−56q360−58q358−59q356 + 18q354 + 106q352 + 143q350 + 38q348−103q346−176q344−183q342−13q340 + 207q338 + 326q336 + 176q334−108q332−327q330−405q328−157q326 + 232q324 + 516q322 + 422q320 + 45q318−352q316−602q314−417q312 + 50q310 + 508q308 + 601q306 + 322q304−124q302−538q300−570q298−253q296 + 219q294 + 491q292 + 462q290 + 190q288−207q286−421q284−384q282−117q280 + 152q278 + 318q276 + 312q274 + 117q272−106q270−264q268−250q266−127q264 + 75q262 + 229q260 + 240q258 + 105q256−102q254−222q252−213q250−36q248 + 147q246 + 239q244 + 141q242−73q240−216q238−204q236−6q234 + 181q232 + 262q230 + 118q228−139q226−297q224−236q222 + 27q220 + 262q218 + 360q216 + 168q214−165q212−396q210−344q208−43q206 + 271q204 + 463q202 + 310q200−59q198−400q196−459q194−225q192 + 129q190 + 456q188 + 453q186 + 166q184−233q182−447q180−400q178−137q176 + 257q174 + 447q172 + 372q170 + 62q168−229q166−396q164−354q162−64q160 + 214q158 + 365q156 + 278q154 + 94q152−154q150−324q148−268q146−99q144 + 121q142 + 229q140 + 261q138 + 127q136−76q134−194q132−221q130−122q128 + 156q124 + 192q122 + 121q120 + 14q118−99q116−142q114−135q112−24q110 + 62q108 + 106q106 + 101q104 + 43q102−23q100−86q98−69q96−38q94 + 7q92 + 44q90 + 55q88 + 36q86−7q84−19q82−31q80−24q78−8q76 + 13q74 + 20q72 + 8q70 + 7q68−3q66−8q64−9q62−q60 + 4q58 + q56 + 5q54 + 3q52 + q50−2q48 + 2q44 + q40 + q38 + q36 + q30

A2 Invariants.

Weight Invariant
1,0 −q36−q32−q30−q26 + 2q24 + q20 + q18 + 2q14 + q10
1,1 q100−2q98 + 4q96−6q94 + 11q92−14q90 + 18q88−24q86 + 28q84−30q82 + 28q80−30q78 + 27q76−16q74 + 8q72 + 4q70−20q68 + 36q66−52q64 + 64q62−70q60 + 72q58−68q56 + 54q54−47q52 + 22q50−10q48−10q46 + 21q44−34q42 + 42q40−38q38 + 38q36−28q34 + 28q32−12q30 + 12q28−4q26 + 4q24 + q20
2,0 q90 + q86 + q84 + q82−2q74−2q72 + q70−3q66 + q62−2q60−4q58−q56−q54−3q52 + 3q48 + 3q42 + q40−q38 + 2q36 + 3q34 + q32 + 3q28 + 2q26 + q20

A3 Invariants.

Weight Invariant
0,1,0 q82−q80 + 3q76−2q74−2q72 + 5q70−2q68−3q66 + 4q64−q62−3q60 + q58−q56−3q54−3q52−5q46 + q44 + 3q42−3q40 + 2q38 + 5q36−q34 + 3q32 + 3q30 + 2q26 + 2q24 + q20
1,0,0 −q47−2q43−2q39−q35 + q33 + q31 + q29 + 2q27 + 2q23 + 2q19 + q15
1,0,1 q132−2q130 + 3q128−q126−3q124 + 9q122−9q120 + 5q118 + 5q116−17q114 + 19q112−11q110−6q108 + 19q106−26q104 + 16q102 + 5q100−18q98 + 31q96−23q94 + 3q92 + 14q90−31q88 + 27q86−8q84 + 2q82 + 17q80−2q78−4q76 + 3q74−14q72−19q70 + 10q68−39q66 + 21q64−20q62−8q60 + 22q58−30q56 + 32q54−11q52 + 9q50 + 16q48−5q46 + 20q44−2q42 + 6q40 + 4q38 + 4q34 + q30

A4 Invariants.

Weight Invariant
0,1,0,0 q104−q100 + 2q98 + 3q96−q94 + 3q90 + q88−2q86 + q84 + 2q82−3q80−4q78−q76−5q74−9q72−2q70−2q68−6q66−q64 + 3q62−2q58 + 3q56 + 3q54 + q52 + 2q50 + 5q48 + 3q46 + 2q44 + 5q42 + 2q40 + 2q38 + 2q36 + 2q34 + q30
1,0,0,0 −q58−2q54−q52−q50−2q48−q44 + q42 + 2q38 + q36 + 2q34 + q32 + q30 + 2q28 + 2q24 + q20

B2 Invariants.

Weight Invariant
0,1 −q82 + q80−2q78 + 3q76−4q74 + 4q72−5q70 + 4q68−3q66 + 2q64 + q62−3q60 + 5q58−7q56 + 7q54−9q52 + 8q50−8q48 + 5q46−3q44 + q42 + q40−2q38 + 5q36−3q34 + 5q32−3q30 + 4q28−2q26 + 2q24 + q20
1,0 q132−q128−q126 + q124 + 3q122 + q120−3q118−3q116 + q114 + 5q112 + q110−4q108−3q106 + 2q104 + 4q102−q100−4q98−q96 + 3q94 + q92−4q90−3q88 + q86 + 2q84−2q82−3q80 + 2q76−q74−4q72−q70 + 4q68 + 3q66−3q64−4q62 + 2q60 + 6q58 + 2q56−2q54−3q52 + 3q50 + 4q48 + q46−2q44 + 2q40 + 2q38 + q30

D4 Invariants.

Weight Invariant
1,0,0,0 q114−q112 + q110−q108 + 3q106−3q104 + 2q102−3q100 + 5q98−3q96 + 2q94−3q92 + 2q90−q86 + q84−4q82 + 4q80−6q78 + 4q76−9q74 + 4q72−8q70 + 4q68−7q66 + 3q64−3q62 + 2q60 + 4q54−q52 + 4q50−q48 + 6q46−q44 + 5q42−q40 + 4q38 + 2q34 + q30

G2 Invariants.

Weight Invariant
1,0 q196−q194 + 2q192−2q190 + q188−2q184 + 5q182−6q180 + 6q178−6q176 + 2q174 + 3q172−7q170 + 12q168−11q166 + 9q164−5q162−2q160 + 6q158−11q156 + 11q154−7q152 + 3q148−7q146 + 5q144−q142−8q140 + 9q138−12q136 + 5q134 + 5q132−15q130 + 20q128−18q126 + 10q124 + q122−12q120 + 18q118−18q116 + 14q114−4q112−4q110 + 11q108−10q106 + 7q104−2q102−5q100 + 8q98−8q96 + 2q94 + 6q92−11q90 + 16q88−11q86 + q84 + 7q82−11q80 + 16q78−12q76 + 6q74 + 2q72−5q70 + 10q68−7q66 + 5q64 + 2q58−2q56 + 2q54 + q50

.

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 9"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 2t3−4t2 + 6t−7 + 6t−1−4t−2 + 2t−3
In[5]:= Conway[K][z]
Out[5]= 2z6 + 8z4 + 8z2 + 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]= {1}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 31, -6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q−3−q−4 + 3q−5−4q−6 + 5q−7−5q−8 + 5q−9−4q−10 + 2q−11−q−12
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z4a10−3z2a10−2a10 + z6a8 + 4z4a8 + 4z2a8 + a8 + z6a6 + 5z4a6 + 7z2a6 + 2a6
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z3a15−za15 + 2z4a14−z2a14 + 3z5a13−3z3a13 + 2za13 + 3z6a12−4z4a12 + 3z2a12 + 2z7a11−2z5a11 + z8a10−z6a10 + 2z4a10−6z2a10 + 2a10 + 3z7a9−8z5a9 + 5z3a9−2za9 + z8a8−3z6a8 + 3z4a8−3z2a8 + a8 + z7a7−3z5a7 + z3a7 + za7 + z6a6−5z4a6 + 7z2a6−2a6

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["9 9"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { 2t3−4t2 + 6t−7 + 6t−1−4t−2 + 2t−3, q−3−q−4 + 3q−5−4q−6 + 5q−7−5q−8 + 5q−9−4q−10 + 2q−11−q−12 }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

[edit] Vassiliev invariants

V2 and V3: (8, -22)
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
32 −176 512 \frac{3760}{3} \frac{560}{3} −5632 -\frac{29696}{3} -\frac{5216}{3} −1264 \frac{16384}{3} 15488 \frac{120320}{3} \frac{17920}{3} \frac{1196284}{15} \frac{40544}{15} \frac{1354096}{45} \frac{4628}{9} \frac{57964}{15}

V2,1 through V6,9 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.

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = -6 is the signature of 9 9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -9-8-7-6-5-4-3-2-10χ
-5         11
-7        110
-9       2  2
-11      21  -1
-13     32   1
-15    22    0
-17   33     0
-19  12      1
-21 13       -2
-23 1        1
-251         -1
Integral Khovanov Homology

(db, data source)

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

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the n + 1-dimensional representation of sl(2))

 n  Jn
2 q−6−q−7 + 4q−9−3q−10−4q−11 + 10q−12−4q−13−11q−14 + 17q−15−2q−16−19q−17 + 21q−18 + 2q−19−25q−20 + 20q−21 + 6q−22−25q−23 + 17q−24 + 6q−25−18q−26 + 10q−27 + 3q−28−8q−29 + 4q−30 + q−31−2q−32 + q−33
3 q−9−q−10 + q−12 + 3q−13−3q−14−3q−15 + q−16 + 10q−17−3q−18−11q−19−5q−20 + 20q−21 + 6q−22−18q−23−18q−24 + 24q−25 + 22q−26−19q−27−33q−28 + 19q−29 + 36q−30−11q−31−45q−32 + 8q−33 + 46q−34 + q−35−51q−36−7q−37 + 51q−38 + 15q−39−53q−40−18q−41 + 48q−42 + 22q−43−44q−44−21q−45 + 37q−46 + 18q−47−28q−48−15q−49 + 23q−50 + 7q−51−13q−52−6q−53 + 11q−54 + q−55−6q−56−q−57 + 5q−58−q−59−q−60−q−61 + 2q−62−q−63
4 q−12−q−13 + q−15 + 3q−17−4q−18−2q−19 + 3q−20 + 11q−22−8q−23−10q−24−q−25−2q−26 + 30q−27−3q−28−16q−29−16q−30−21q−31 + 51q−32 + 15q−33−3q−34−28q−35−61q−36 + 54q−37 + 28q−38 + 31q−39−17q−40−102q−41 + 38q−42 + 17q−43 + 65q−44 + 19q−45−121q−46 + 15q−47−17q−48 + 85q−49 + 63q−50−119q−51−q−52−61q−53 + 91q−54 + 103q−55−103q−56−16q−57−104q−58 + 94q−59 + 137q−60−82q−61−30q−62−138q−63 + 87q−64 + 158q−65−54q−66−32q−67−157q−68 + 65q−69 + 152q−70−27q−71−12q−72−144q−73 + 33q−74 + 113q−75−14q−76 + 14q−77−100q−78 + 12q−79 + 60q−80−17q−81 + 27q−82−50q−83 + 6q−84 + 24q−85−20q−86 + 21q−87−19q−88 + 7q−89 + 8q−90−16q−91 + 11q−92−6q−93 + 4q−94 + 3q−95−7q−96 + 4q−97−2q−98 + q−99 + q−100−2q−101 + q−102
5 q−15−q−16 + q−18 + 2q−21−3q−22−2q−23 + 3q−24 + 3q−25 + 5q−27−7q−28−10q−29−q−30 + 7q−31 + 8q−32 + 18q−33−4q−34−23q−35−20q−36−7q−37 + 12q−38 + 46q−39 + 25q−40−15q−41−40q−42−49q−43−22q−44 + 55q−45 + 70q−46 + 33q−47−17q−48−79q−49−89q−50 + 4q−51 + 77q−52 + 90q−53 + 56q−54−45q−55−125q−56−80q−57 + 10q−58 + 92q−59 + 126q−60 + 47q−61−80q−62−128q−63−95q−64 + 15q−65 + 139q−66 + 143q−67 + 26q−68−104q−69−178q−70−103q−71 + 87q−72 + 201q−73 + 146q−74−29q−75−218q−76−215q−77 + 3q−78 + 225q−79 + 253q−80 + 47q−81−231q−82−308q−83−74q−84 + 240q−85 + 340q−86 + 115q−87−246q−88−388q−89−139q−90 + 251q−91 + 416q−92 + 181q−93−248q−94−453q−95−207q−96 + 233q−97 + 458q−98 + 249q−99−200q−100−458q−101−275q−102 + 158q−103 + 424q−104 + 288q−105−98q−106−372q−107−288q−108 + 48q−109 + 305q−110 + 254q−111−3q−112−218q−113−221q−114−29q−115 + 160q−116 + 157q−117 + 37q−118−83q−119−118q−120−40q−121 + 53q−122 + 65q−123 + 29q−124−15q−125−40q−126−20q−127 + 8q−128 + 11q−129 + 13q−130 + 6q−131−9q−132−5q−133−5q−135 + q−136 + 10q−137−4q−138−q−139 + 3q−140−5q−141−q−142 + 5q−143−2q−144−q−145 + 2q−146−q−147−q−148 + 2q−149−q−150
6 q−18−q−19 + q−21−q−24 + 3q−25−3q−26−2q−27 + 4q−28 + 2q−29 + 2q−30−5q−31 + 6q−32−8q−33−10q−34 + 5q−35 + 7q−36 + 12q−37−4q−38 + 18q−39−15q−40−31q−41−11q−42 + 24q−44 + 8q−45 + 61q−46 + 4q−47−42q−48−48q−49−45q−50−7q−51−9q−52 + 124q−53 + 70q−54 + 16q−55−43q−56−89q−57−93q−58−120q−59 + 117q−60 + 113q−61 + 129q−62 + 64q−63−18q−64−129q−65−276q−66−5q−67 + 14q−68 + 156q−69 + 182q−70 + 185q−71 + 5q−72−308q−73−113q−74−206q−75−13q−76 + 126q−77 + 355q−78 + 253q−79−124q−80−26q−81−357q−82−291q−83−164q−84 + 313q−85 + 420q−86 + 167q−87 + 284q−88−282q−89−481q−90−558q−91 + 50q−92 + 373q−93 + 381q−94 + 683q−95 + 5q−96−478q−97−885q−98−304q−99 + 139q−100 + 440q−101 + 1024q−102 + 374q−103−325q−104−1077q−105−618q−106−162q−107 + 388q−108 + 1255q−109 + 707q−110−133q−111−1175q−112−853q−113−425q−114 + 327q−115 + 1413q−116 + 964q−117 + 11q−118−1256q−119−1040q−120−623q−121 + 315q−122 + 1556q−123 + 1178q−124 + 109q−125−1348q−126−1223q−127−800q−128 + 306q−129 + 1676q−130 + 1385q−131 + 244q−132−1359q−133−1377q−134−1002q−135 + 183q−136 + 1662q−137 + 1543q−138 + 467q−139−1164q−140−1371q−141−1168q−142−86q−143 + 1395q−144 + 1506q−145 + 681q−146−767q−147−1099q−148−1139q−149−358q−150 + 923q−151 + 1189q−152 + 713q−153−353q−154−653q−155−860q−156−451q−157 + 465q−158 + 722q−159 + 528q−160−103q−161−256q−162−483q−163−357q−164 + 181q−165 + 333q−166 + 280q−167−25q−168−37q−169−199q−170−207q−171 + 62q−172 + 113q−173 + 110q−174−18q−175 + 36q−176−58q−177−102q−178 + 25q−179 + 25q−180 + 33q−181−17q−182 + 38q−183−8q−184−46q−185 + 12q−186 + q−187 + 7q−188−12q−189 + 22q−190 + 2q−191−18q−192 + 7q−193−2q−194 + 2q−195−7q−196 + 8q−197 + 2q−198−7q−199 + 4q−200−q−201 + q−202−2q−203 + q−204 + q−205−2q−206 + q−207

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

[edit] Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

9_8

9_10

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