9 3

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9_2

9_4

Contents

Image:9 3.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

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

Visit 9_3's page at Knotilus!

Visit 9 3's page at the original Knot Atlas!

[edit 9 3 Quick Notes]
[edit 9 3 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X14,6,15,5 X16,8,17,7 X2,12,3,11 X4,14,5,13 X6,16,7,15
Gauss code 1, -7, 2, -8, 5, -9, 6, -1, 3, -4, 7, -2, 8, -5, 9, -6, 4, -3
Dowker-Thistlethwaite code 8 12 14 16 18 2 4 6 10
Conway Notation [63]


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

Length is 10, width is 3,

Braid index is 3

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

[edit Notes on presentations of 9 3]


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 3"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X14,6,15,5 X16,8,17,7 X2,12,3,11 X4,14,5,13 X6,16,7,15
In[5]:= GaussCode[K]
Out[5]= 1, -7, 2, -8, 5, -9, 6, -1, 3, -4, 7, -2, 8, -5, 9, -6, 4, -3
In[6]:= DTCode[K]
Out[6]= 8 12 14 16 18 2 4 6 10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [63]
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,{1,1,1,1,1,1,1,2,−1,2})
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:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.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 3_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{4, 11}, {3, 5}, {6, 4}, {5, 10}, {2, 6}, {11, 9}, {10, 8}, {9, 7}, {1, 3}, {8, 2}, {7, 1}]
In[14]:= Draw[ap]
Image:9 3_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 [5][-16]
Hyperbolic Volume 4.99486
A-Polynomial See Data:9 3/A-polynomial

[edit Notes for 9 3'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 3's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial 2t3−3t2 + 3t−3 + 3t−1−3t−2 + 2t−3
Conway polynomial 2z6 + 9z4 + 9z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 19, 6 }
Jones polynomial −q12 + q11−2q10 + 3q9−3q8 + 3q7−2q6 + 2q5−q4 + q3
HOMFLY-PT polynomial (db, data sources) z6a−6 + z6a−8 + 5z4a−6 + 5z4a−8−z4a−10 + 6z2a−6 + 7z2a−8−4z2a−10 + a−6 + 3a−8−3a−10
Kauffman polynomial (db, data sources) z8a−8 + z8a−10 + z7a−7 + 2z7a−9 + z7a−11 + z6a−6−5z6a−8−5z6a−10 + z6a−12−4z5a−7−8z5a−9−3z5a−11 + z5a−13−5z4a−6 + 9z4a−8 + 11z4a−10−2z4a−12 + z4a−14 + 3z3a−7 + 9z3a−9 + 4z3a−11−z3a−13 + z3a−15 + 6z2a−6−9z2a−8−11z2a−10 + 3z2a−12−z2a−14−4za−9−za−11 + za−13−2za−15−a−6 + 3a−8 + 3a−10
The A2 invariant q−10 + q−14 + q−18 + q−20 + q−22 + 2q−24−q−30−q−32−q−34−q−36
The G2 invariant q−50 + q−54−q−56 + q−58 + 3q−64−2q−66 + 3q−68−q−70 + q−72 + 2q−74−3q−76 + 3q−78−q−80 + q−82 + q−84−q−86 + 2q−88 + q−90 + 2q−94−q−96 + q−98 + 2q−100−2q−102 + 3q−104−q−106 + 2q−108−q−112 + 2q−114−2q−116 + 3q−118−2q−120 + q−124−2q−126 + q−128−q−130−q−132 + q−134−q−136−q−138−q−142−q−146−2q−148−q−152−q−156−q−160−q−162−q−164−q−166 + 2q−168−2q−170 + q−172 + q−178−q−180 + q−182−q−184 + q−186−q−190 + q−192 + q−196
Further Quantum Invariants
Further quantum knot invariants for 9_3.

A1 Invariants.

Weight Invariant
1 q−5 + q−9 + q−13 + q−19−q−21−q−25
2 q−10 + 2q−16 + q−18−q−20 + q−22 + q−24−q−26 + q−30 + q−36−q−40−q−48 + q−50−q−52−2q−54−q−58 + q−62 + q−68
3 q−15 + q−21 + 2q−23 + q−25−q−27−q−29 + 2q−31 + 2q−33−2q−37 + 2q−41 + q−43−q−45−q−47 + q−51 + q−53−q−57 + 2q−61−2q−65−q−67 + q−69−q−71−2q−73 + q−77−q−81−q−83−q−85 + q−87 + q−89−2q−91−q−93−q−95 + 2q−97−q−101−q−103 + q−105 + 2q−107 + q−109−q−111 + 2q−115 + q−117−q−121−q−129
4 q−20 + q−26 + q−28 + 2q−30−q−34 + 4q−40 + 2q−42−q−44−2q−46−3q−48 + 2q−50 + 3q−52 + 2q−54−4q−58−q−60 + q−62 + 2q−64 + 2q−66−q−68−q−72−q−74 + q−76 + q−78 + 3q−80−4q−84−3q−86 + 4q−90 + 2q−92−4q−94−4q−96−q−98 + 4q−100 + 3q−102−3q−104−4q−106−q−108 + 2q−110 + 2q−112−2q−114−2q−116 + q−120 + q−122−2q−124−2q−126 + q−130 + q−132 + q−134−q−136−3q−138−2q−140 + 2q−142 + 6q−144 + 3q−146−q−148−7q−150−3q−152 + 6q−154 + 6q−156 + 2q−158−7q−160−4q−162 + 3q−164 + 5q−166 + 4q−168−3q−170−4q−172 + q−174 + 2q−176 + 3q−178−2q−180−3q−182−q−184 + 2q−188−q−190−q−192−q−194−q−196 + q−198 + q−208
5 q−25 + q−31 + q−33 + q−35 + q−37−q−41 + 2q−45 + 2q−47 + 3q−49 + q−51−2q−53−4q−55−2q−57 + q−59 + 4q−61 + 5q−63 + 2q−65−2q−67−5q−69−4q−71 + 3q−75 + 5q−77 + 3q−79−q−81−4q−83−3q−85 + q−89 + 2q−91 + 2q−93 + q−95 + q−97 + q−99−2q−101−4q−103−4q−105 + 5q−109 + 6q−111 + 2q−113−5q−115−10q−117−6q−119 + 3q−121 + 9q−123 + 7q−125−q−127−9q−129−10q−131−3q−133 + 7q−135 + 10q−137 + 4q−139−4q−141−9q−143−6q−145 + 2q−147 + 8q−149 + 5q−151−3q−153−7q−155−7q−157 + 6q−161 + 5q−163−4q−167−3q−169 + q−171 + 3q−173 + 3q−175−2q−177−3q−179 + 2q−183 + 2q−185 + q−187−q−189−2q−191−2q−193 + 2q−197 + 4q−199 + 6q−201 + 2q−203−5q−205−8q−207−4q−209 + 3q−211 + 11q−213 + 13q−215−q−217−12q−219−13q−221−3q−223 + 9q−225 + 15q−227 + 5q−229−8q−231−11q−233−6q−235 + 5q−237 + 8q−239 + 3q−241−4q−243−6q−245−3q−247 + 3q−249 + 4q−251 + q−253−4q−255−3q−257−q−259 + 2q−261 + 2q−263 + q−265−2q−267−3q−269−q−271 + q−273 + q−275 + 2q−277 + q−279−q−281 + q−289 + q−291−q−305
6 q−30 + q−36 + q−38 + q−40 + q−44−q−48 + q−50 + 2q−52 + 3q−54 + q−56 + 2q−58−q−60−4q−62−3q−64−q−66 + 3q−68 + 3q−70 + 7q−72 + 4q−74−2q−76−5q−78−7q−80−4q−82−2q−84 + 6q−86 + 8q−88 + 6q−90 + 2q−92−3q−94−6q−96−9q−98−2q−100 + 2q−102 + 6q−104 + 6q−106 + 3q−108 + q−110−4q−112−2q−114−2q−116−q−120−2q−122 + 5q−128 + 6q−130 + 5q−132−3q−134−11q−136−11q−138−9q−140 + 2q−142 + 12q−144 + 17q−146 + 9q−148−5q−150−15q−152−20q−154−11q−156 + 4q−158 + 18q−160 + 19q−162 + 9q−164−3q−166−18q−168−20q−170−11q−172 + 5q−174 + 16q−176 + 17q−178 + 12q−180−5q−182−18q−184−19q−186−9q−188 + 5q−190 + 16q−192 + 20q−194 + 7q−196−10q−198−19q−200−15q−202−3q−204 + 10q−206 + 20q−208 + 11q−210−5q−212−14q−214−12q−216−3q−218 + 7q−220 + 16q−222 + 10q−224−4q−226−10q−228−8q−230−2q−232 + 4q−234 + 8q−236 + 4q−238−4q−240−4q−242 + 2q−246 + 3q−248 + 3q−250−3q−254−3q−256 + 3q−260 + 4q−262 + 4q−264 + 2q−266 + q−268−4q−270−8q−272−7q−274−2q−276 + 4q−278 + 12q−280 + 15q−282 + 6q−284−9q−286−18q−288−17q−290−10q−292 + 12q−294 + 25q−296 + 22q−298 + 5q−300−15q−302−24q−304−25q−306−3q−308 + 15q−310 + 22q−312 + 13q−314−q−316−12q−318−18q−320−7q−322 + 7q−326 + 7q−328 + 4q−330 + 3q−332−3q−334−3q−338−4q−340−4q−342 + 6q−346 + 3q−348 + 6q−350−4q−354−6q−356−3q−358 + 2q−360 + 2q−362 + 6q−364 + 3q−366−3q−370−3q−372−q−374−q−376 + 3q−378 + 2q−380 + 2q−382−q−388−2q−390−q−394 + q−396−q−404−q−408 + q−420

A2 Invariants.

Weight Invariant
1,0 q−10 + q−14 + q−18 + q−20 + q−22 + 2q−24−q−30−q−32−q−34−q−36
1,1 q−20 + 2q−24−2q−26 + 6q−28−2q−30 + 10q−32−4q−34 + 9q−36−2q−38 + 4q−40−4q−44 + 6q−46−8q−48 + 8q−50−11q−52 + 8q−54−10q−56 + 4q−58−8q−60−2q−64−2q−66 + 2q−68 + 2q−72 + 2q−74 + q−76−2q−78−2q−82 + 3q−84−4q−86 + 2q−88−2q−90 + 2q−92−2q−94 + 2q−96 + q−100
2,0 q−20 + q−26 + 2q−28 + q−30 + q−32 + 2q−34 + 2q−36 + q−42 + q−44 + q−46 + 2q−48 + q−50−q−58−3q−66−3q−68−3q−70−3q−72−3q−74−2q−76 + q−80 + 2q−82 + q−84 + 2q−86 + q−88 + q−90

A3 Invariants.

Weight Invariant
0,1,0 q−20 + q−24 + q−26 + q−30 + 2q−32 + 2q−34 + 4q−36 + 3q−38 + 2q−40 + 2q−42 + q−44−2q−46−q−48−2q−50−3q−52−3q−54−2q−56−q−58−q−60 + q−64−q−66−q−68 + q−70−q−72−q−74 + q−76 + q−78 + q−82
1,0,0 q−15 + q−19 + q−23 + 2q−27 + 2q−29 + 2q−31 + 2q−33−2q−39−q−41−2q−43−q−45−q−47
1,0,1 q−30 + 2q−34 + q−38 + 3q−40 + 2q−42 + 9q−44 + 5q−46 + 8q−48 + 7q−50 + 5q−52 + 7q−54 + 2q−56 + 5q−58−q−60 + 4q−62−4q−64−2q−66−7q−68−12q−70−7q−72−14q−74−4q−76−9q−78 + q−80−3q−82 + 2q−84 + 3q−86−2q−88 + 6q−90−2q−92 + 6q−94 + q−96 + 3q−98 + 2q−100−q−102−2q−104−q−106 + q−108−2q−110 + 3q−112−2q−114−q−116 + q−118−2q−120 + q−122−q−124 + 2q−128 + q−132

A4 Invariants.

Weight Invariant
0,1,0,0 q−30 + q−34 + q−36 + q−38 + 2q−42 + 2q−44 + 3q−46 + 4q−48 + 4q−50 + 5q−52 + 5q−54 + 4q−56 + 3q−58 + 3q−60 + 2q−62−q−64−3q−66−4q−68−5q−70−7q−72−5q−74−4q−76−4q−78−2q−80−q−82−2q−84−2q−86 + q−94 + 3q−96 + 2q−98 + q−100 + q−102 + q−104
1,0,0,0 q−20 + q−24 + q−28 + q−32 + 2q−34 + 2q−36 + 3q−38 + 2q−40 + 2q−42−2q−48−2q−50−2q−52−2q−54−q−56−q−58

B2 Invariants.

Weight Invariant
0,1 q−20 + q−24−q−26 + 2q−28−q−30 + 2q−32 + 2q−36 + q−38 + 2q−42−q−44 + 2q−46−3q−48 + 2q−50−3q−52 + q−54−2q−56 + q−58−q−60 + q−64−q−66 + q−68−q−70 + q−72−q−74 + q−76−q−78−q−82
1,0 q−30 + q−38 + q−40−q−44 + q−46 + 2q−48 + q−50−q−52 + q−54 + 2q−56 + 3q−58 + q−60 + 2q−66 + q−68−q−72−q−78−q−80−q−82−q−86−2q−88−2q−90−q−96−q−98 + q−102−q−106−q−108 + q−112−q−116−q−118 + q−122 + q−124 + q−132

D4 Invariants.

Weight Invariant
1,0,0,0 q−30 + q−34 + 2q−38−q−40 + 2q−42 + 3q−46 + 2q−48 + 3q−50 + 3q−52 + 4q−54 + 4q−56 + 2q−58 + 3q−60−q−62 + q−64−4q−66−q−68−5q−70−2q−72−5q−74−q−76−3q−78−q−82 + q−90−q−92−q−96 + q−98−q−100−q−104 + q−106 + q−110 + q−114

G2 Invariants.

Weight Invariant
1,0 q−50 + q−54−q−56 + q−58 + 3q−64−2q−66 + 3q−68−q−70 + q−72 + 2q−74−3q−76 + 3q−78−q−80 + q−82 + q−84−q−86 + 2q−88 + q−90 + 2q−94−q−96 + q−98 + 2q−100−2q−102 + 3q−104−q−106 + 2q−108−q−112 + 2q−114−2q−116 + 3q−118−2q−120 + q−124−2q−126 + q−128−q−130−q−132 + q−134−q−136−q−138−q−142−q−146−2q−148−q−152−q−156−q−160−q−162−q−164−q−166 + 2q−168−2q−170 + q−172 + q−178−q−180 + q−182−q−184 + q−186−q−190 + q−192 + q−196

.

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 3"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 2t3−3t2 + 3t−3 + 3t−1−3t−2 + 2t−3
In[5]:= Conway[K][z]
Out[5]= 2z6 + 9z4 + 9z2 + 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]= { 19, 6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q12 + q11−2q10 + 3q9−3q8 + 3q7−2q6 + 2q5−q4 + q3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z6a−6 + z6a−8 + 5z4a−6 + 5z4a−8−z4a−10 + 6z2a−6 + 7z2a−8−4z2a−10 + a−6 + 3a−8−3a−10
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z8a−8 + z8a−10 + z7a−7 + 2z7a−9 + z7a−11 + z6a−6−5z6a−8−5z6a−10 + z6a−12−4z5a−7−8z5a−9−3z5a−11 + z5a−13−5z4a−6 + 9z4a−8 + 11z4a−10−2z4a−12 + z4a−14 + 3z3a−7 + 9z3a−9 + 4z3a−11−z3a−13 + z3a−15 + 6z2a−6−9z2a−8−11z2a−10 + 3z2a−12−z2a−14−4za−9−za−11 + za−13−2za−15−a−6 + 3a−8 + 3a−10

[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 3"];
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]= { 2t3−3t2 + 3t−3 + 3t−1−3t−2 + 2t−3, −q12 + q11−2q10 + 3q9−3q8 + 3q7−2q6 + 2q5−q4 + q3 }
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: (9, 26)

[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 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 0123456789χ
25         1-1
23          0
21       21 -1
19      1   1
17     22   0
15    11    0
13   12     1
11  11      0
9  1       1
711        0
51         1
Integral Khovanov Homology

(db, data source)

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

[edit] The Coloured Jones Polynomials

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

 n  Jn
2 q33−q32 + 2q30−2q29−q28 + 3q27−4q26 + 5q24−6q23 + q22 + 5q21−6q20 + 6q18−5q17−q16 + 6q15−4q14−2q13 + 5q12−2q11−2q10 + 3q9−q7 + q6
3 −q63 + q62−2q59 + 2q58 + q57 + q56−4q55 + q54 + 3q53 + 2q52−5q51−q50 + 3q49 + 3q48−3q47−4q46 + 3q45 + 2q44−4q42 + q41 + 2q40−3q38 + 2q37 + q36−2q35−2q34 + 4q33−q32−3q31 + 6q29−3q28−4q27 + q26 + 7q25−3q24−5q23 + 7q21−q20−4q19−2q18 + 5q17 + q16−2q15−2q14 + 2q13 + q12−q10 + q9
4 q102−q101 + 2q97−3q96 + 5q92−5q91−q90−2q89 + q88 + 10q87−6q86−2q85−7q84 + 2q83 + 17q82−5q81−4q80−14q79−q78 + 26q77−q76−4q75−23q74−5q73 + 32q72 + 3q71−q70−27q69−9q68 + 31q67 + 5q66 + q65−27q64−9q63 + 30q62 + 3q61 + q60−24q59−9q58 + 29q57 + q56 + q55−20q54−9q53 + 26q52−2q51 + 2q50−14q49−8q48 + 21q47−5q46 + 2q45−8q44−6q43 + 17q42−8q41 + q40−4q39−3q38 + 15q37−8q36−q35−4q34−2q33 + 14q32−5q31−q30−5q29−4q28 + 11q27−q26 + q25−4q24−5q23 + 6q22 + 2q20−q19−3q18 + 2q17 + q15−q13 + q12
5 −q150 + q149−q144 + 2q143−q141−2q138 + 4q137 + q136−2q135−2q133−4q132 + 5q131 + 4q130−q129−5q127−6q126 + 4q125 + 9q124 + 2q123−q122−11q121−9q120 + 6q119 + 16q118 + 7q117−4q116−22q115−14q114 + 9q113 + 29q112 + 17q111−10q110−34q109−24q108 + 10q107 + 40q106 + 31q105−12q104−42q103−31q102 + 6q101 + 43q100 + 38q99−8q98−44q97−33q96 + 4q95 + 41q94 + 38q93−7q92−42q91−32q90 + 4q89 + 39q88 + 35q87−6q86−37q85−32q84 + 2q83 + 35q82 + 34q81−2q80−32q79−31q78−4q77 + 28q76 + 34q75 + 2q74−24q73−28q72−10q71 + 20q70 + 31q69 + 7q68−16q67−22q66−13q65 + 10q64 + 24q63 + 8q62−8q61−14q60−11q59 + 4q58 + 15q57 + 4q56−3q55−7q54−7q53 + 3q52 + 10q51−3q49−5q48−4q47 + 3q46 + 10q45 + q44−3q43−6q42−5q41 + 9q39 + 4q38 + q37−4q36−7q35−3q34 + 5q33 + 3q32 + 4q31−4q29−4q28 + 2q27 + 2q25 + 2q24−q23−2q22 + q21 + q18−q16 + q15
6 q207−q206−q201 + 2q200−2q199 + q198 + q195−3q194 + 3q193−4q192 + 2q191 + q190 + 3q188−3q187 + 4q186−8q185 + 2q184−q183 + 6q181 + 7q179−12q178 + 2q177−6q176−3q175 + 8q174 + 4q173 + 13q172−15q171 + 5q170−12q169−7q168 + 8q167 + 5q166 + 16q165−18q164 + 11q163−11q162−4q161 + 8q160−2q159 + 9q158−29q157 + 17q156 + 10q154 + 17q153−9q152−9q151−51q150 + 18q149 + 9q148 + 30q147 + 34q146−6q145−22q144−73q143 + 11q142 + 8q141 + 39q140 + 49q139 + 3q138−25q137−81q136 + 5q135 + 3q134 + 38q133 + 53q132 + 8q131−24q130−79q129 + 5q128 + 2q127 + 35q126 + 50q125 + 8q124−21q123−76q122 + 5q121 + q120 + 33q119 + 46q118 + 8q117−13q116−72q115 + q114−5q113 + 27q112 + 44q111 + 14q110 + q109−66q108−8q107−15q106 + 18q105 + 42q104 + 23q103 + 17q102−57q101−18q100−28q99 + 6q98 + 38q97 + 32q96 + 34q95−44q94−22q93−39q92−8q91 + 28q90 + 33q89 + 47q88−27q87−17q86−40q85−19q84 + 12q83 + 24q82 + 49q81−12q80−5q79−30q78−20q77−2q76 + 9q75 + 40q74−7q73 + 5q72−16q71−12q70−7q69−q68 + 29q67−9q66 + 5q65−8q64−4q63−6q62−2q61 + 24q60−9q59 + 3q58−7q57−3q56−8q55−2q54 + 22q53−4q52 + 5q51−4q50−3q49−12q48−6q47 + 15q46−q45 + 8q44 + q43 + q42−10q41−8q40 + 7q39−3q38 + 5q37 + 3q36 + 4q35−4q34−5q33 + 3q32−3q31 + q30 + q29 + 3q28−q27−2q26 + 2q25−q24 + q21−q19 + q18

[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_2

9_4

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