9 36

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9_35

9_37

Contents

Image:9 36.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 36's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9_36's page at Knotilus!

Visit 9 36's page at the original Knot Atlas!

[edit 9 36 Quick Notes]
[edit 9 36 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X7,10,8,11 X3948 X9,3,10,2 X11,17,12,16 X5,15,6,14 X15,7,16,6 X13,1,14,18 X17,13,18,12
Gauss code -1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -5, 9, -8, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code 4 8 14 10 2 16 18 6 12
Conway Notation [22,3,2]


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

Length is 9, width is 4,

Braid index is 4

Image:9 36_ML.gif Image:9 36_AP.gif
[{11, 5}, {6, 4}, {5, 10}, {3, 6}, {8, 11}, {7, 9}, {4, 8}, {2, 7}, {1, 3}, {10, 2}, {9, 1}]

[edit Notes on presentations of 9 36]


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 36"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X7,10,8,11 X3948 X9,3,10,2 X11,17,12,16 X5,15,6,14 X15,7,16,6 X13,1,14,18 X17,13,18,12
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -5, 9, -8, 6, -7, 5, -9, 8
In[6]:= DTCode[K]
Out[6]= 4 8 14 10 2 16 18 6 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [22,3,2]
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(4,{1,1,1,−2,1,1,3,−2,3})
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]= { 4, 9, 4 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.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 36_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{11, 5}, {6, 4}, {5, 10}, {3, 6}, {8, 11}, {7, 9}, {4, 8}, {2, 7}, {1, 3}, {10, 2}, {9, 1}]
In[14]:= Draw[ap]
Image:9 36_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index {4,7}
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-12]
Hyperbolic Volume 9.88458
A-Polynomial See Data:9 36/A-polynomial

[edit Notes for 9 36's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 2
Topological 4 genus 2
Concordance genus 3
Rasmussen s-Invariant 4

[edit Notes for 9 36's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t3 + 5t2−8t + 9−8t−1 + 5t−2−t−3
Conway polynomial −z6−z4 + 3z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 37, 4 }
Jones polynomial −q9 + 2q8−4q7 + 6q6−6q5 + 6q4−5q3 + 4q2−2q + 1
HOMFLY-PT polynomial (db, data sources) −z6a−4 + z4a−2−4z4a−4 + 2z4a−6 + 3z2a−2−5z2a−4 + 6z2a−6−z2a−8 + 2a−2−3a−4 + 4a−6−2a−8
Kauffman polynomial (db, data sources) z8a−4 + z8a−6 + 2z7a−3 + 5z7a−5 + 3z7a−7 + z6a−2 + z6a−4 + 4z6a−6 + 4z6a−8−7z5a−3−14z5a−5−4z5a−7 + 3z5a−9−4z4a−2−12z4a−4−17z4a−6−7z4a−8 + 2z4a−10 + 6z3a−3 + 9z3a−5−2z3a−9 + z3a−11 + 5z2a−2 + 12z2a−4 + 15z2a−6 + 7z2a−8−z2a−10−za−3−2za−5 + za−7 + za−9−za−11−2a−2−3a−4−4a−6−2a−8
The A2 invariant 1 + q−4 + q−6−q−8 + q−10−2q−12 + q−14 + q−16 + q−18 + 2q−20−q−22−q−26−q−28
The G2 invariant q−2−q−4 + 4q−6−5q−8 + 5q−10−2q−12−4q−14 + 14q−16−17q−18 + 19q−20−11q−22−2q−24 + 18q−26−27q−28 + 28q−30−17q−32 + q−34 + 13q−36−23q−38 + 20q−40−9q−42−5q−44 + 15q−46−18q−48 + 9q−50 + 3q−52−18q−54 + 24q−56−24q−58 + 16q−60 + q−62−18q−64 + 32q−66−33q−68 + 28q−70−9q−72−10q−74 + 25q−76−28q−78 + 24q−80−7q−82−7q−84 + 19q−86−16q−88 + 6q−90 + 6q−92−16q−94 + 18q−96−11q−98−2q−100 + 11q−102−18q−104 + 20q−106−15q−108 + 4q−110 + 3q−112−12q−114 + 12q−116−13q−118 + 10q−120−5q−122 + q−124 + 3q−126−7q−128 + 6q−130−5q−132 + 4q−134−2q−136 + q−140−2q−142 + 2q−144−q−146 + q−148
Further Quantum Invariants
Further quantum knot invariants for 9_36.

A1 Invariants.

Weight Invariant
1 q−q−1 + 2q−3−q−5 + q−7 + 2q−13−2q−15 + q−17−q−19
2 q6−q4−2q2 + 4 + q−2−6q−4 + 4q−6 + 5q−8−6q−10 + q−12 + 7q−14−4q−16−3q−18 + 5q−20−4q−24 + q−26 + 4q−28−3q−30−5q−32 + 7q−34−7q−38 + 5q−40 + q−42−4q−44 + 2q−46−q−50 + q−52
3 q15−q13−2q11 + 5q7 + 3q5−7q3−8q + 6q−1 + 14q−3−18q−7−6q−9 + 17q−11 + 16q−13−11q−15−21q−17 + 5q−19 + 23q−21 + 4q−23−21q−25−10q−27 + 20q−29 + 15q−31−16q−33−17q−35 + 12q−37 + 19q−39−9q−41−21q−43 + 3q−45 + 18q−47 + 3q−49−16q−51−12q−53 + 10q−55 + 20q−57−2q−59−24q−61−5q−63 + 22q−65 + 14q−67−20q−69−14q−71 + 12q−73 + 12q−75−6q−77−8q−79 + 3q−81 + 4q−83−2q−85−q−87 + q−89 + q−97−q−99
4 q28−q26−2q24 + q20 + 7q18 + q16−7q14−9q12−8q10 + 17q8 + 19q6 + 5q4−17q2−39−2q−2 + 28q−4 + 44q−6 + 18q−8−51q−10−49q−12−17q−14 + 51q−16 + 78q−18 + 3q−20−54q−22−83q−24−10q−26 + 86q−28 + 73q−30 + 8q−32−94q−34−82q−36 + 29q−38 + 93q−40 + 71q−42−55q−44−104q−46−27q−48 + 70q−50 + 92q−52−15q−54−91q−56−48q−58 + 52q−60 + 85q−62−77q−66−54q−68 + 37q−70 + 75q−72 + 23q−74−56q−76−71q−78−q−80 + 55q−82 + 69q−84 + 4q−86−75q−88−69q−90−6q−92 + 99q−94 + 89q−96−29q−98−105q−100−89q−102 + 60q−104 + 126q−106 + 38q−108−69q−110−113q−112−2q−114 + 81q−116 + 55q−118−8q−120−67q−122−22q−124 + 25q−126 + 26q−128 + 11q−130−22q−132−8q−134 + 4q−136 + 3q−138 + 7q−140−6q−142−q−144 + q−146−q−148 + 3q−150−2q−152−q−158 + q−160
5 q45−q43−2q41 + q37 + 3q35 + 5q33 + q31−9q29−11q27−6q25 + 5q23 + 21q21 + 25q19 + 6q17−27q15−44q13−34q11 + 8q9 + 59q7 + 76q5 + 35q3−45q−106q−1−100q−3−15q−5 + 100q−7 + 164q−9 + 110q−11−40q−13−176q−15−207q−17−84q−19 + 125q−21 + 269q−23 + 221q−25 + 2q−27−242q−29−333q−31−173q−33 + 140q−35 + 375q−37 + 336q−39 + 30q−41−322q−43−450q−45−221q−47 + 200q−49 + 482q−51 + 381q−53−35q−55−433q−57−485q−59−128q−61 + 338q−63 + 519q−65 + 253q−67−222q−69−494q−71−330q−73 + 118q−75 + 436q−77 + 348q−79−47q−81−372q−83−329q−85 + 8q−87 + 313q−89 + 296q−91−3q−93−278q−95−265q−97 + 8q−99 + 260q−101 + 253q−103 + 5q−105−247q−107−268q−109−46q−111 + 216q−113 + 294q−115 + 133q−117−138q−119−318q−121−258q−123 + 15q−125 + 301q−127 + 383q−129 + 166q−131−219q−133−481q−135−364q−137 + 81q−139 + 496q−141 + 531q−143 + 115q−145−430q−147−633q−149−289q−151 + 295q−153 + 616q−155 + 423q−157−124q−159−530q−161−459q−163−22q−165 + 380q−167 + 414q−169 + 112q−171−228q−173−320q−175−139q−177 + 112q−179 + 213q−181 + 116q−183−39q−185−118q−187−82q−189 + 6q−191 + 60q−193 + 48q−195 + q−197−25q−199−22q−201−4q−203 + 9q−205 + 11q−207 + 2q−209−7q−211−3q−213 + 2q−215 + 2q−219 + q−221−3q−223−q−225 + 2q−227 + q−233−q−235

A2 Invariants.

Weight Invariant
1,0 1 + q−4 + q−6−q−8 + q−10−2q−12 + q−14 + q−16 + q−18 + 2q−20−q−22−q−26−q−28
1,1 q4−2q2 + 8−16q−2 + 29q−4−44q−6 + 60q−8−74q−10 + 80q−12−72q−14 + 62q−16−32q−18 + 3q−20 + 38q−22−70q−24 + 100q−26−123q−28 + 132q−30−134q−32 + 118q−34−96q−36 + 64q−38−30q−40 + 28q−44−44q−46 + 52q−48−52q−50 + 46q−52−44q−54 + 34q−56−28q−58 + 24q−60−20q−62 + 16q−64−12q−66 + 9q−68−6q−70 + 4q−72−2q−74 + q−76
2,0 q4−1 + 2q−4 + q−6−2q−8−q−10 + 4q−12 + q−14−2q−16 + 2q−18 + 3q−20−q−24 + 2q−26 + 2q−28−q−30 + 2q−32 + q−34−4q−36−3q−38 + 2q−40−2q−42−3q−44 + 2q−46 + 4q−48 + q−50−2q−52 + q−54−3q−58−2q−60−q−62 + q−66 + q−68 + q−70

A3 Invariants.

Weight Invariant
0,1,0 1−q−2 + 2q−4 + 2q−6−2q−8 + 4q−10 + q−12−4q−14 + 4q−16−q−18−5q−20 + 4q−22 + 2q−24−3q−26 + 3q−28 + 3q−30 + q−32−q−34 + q−36 + 3q−38−5q−40−2q−42 + 4q−44−6q−46−2q−48 + 5q−50−3q−52−2q−54 + 3q−56−q−60 + q−62
1,0,0 q−1 + 2q−5 + 2q−9−q−11−2q−15−q−17 + q−21 + 3q−23 + q−25 + 3q−27−q−29−2q−33−q−35−q−37

A4 Invariants.

Weight Invariant
0,1,0,0 q−2 + q−6 + 3q−8 + q−10 + q−12 + 3q−14−3q−18 + q−22−3q−24−q−26 + 6q−28 + 5q−30−4q−32 + 2q−34 + 5q−36−5q−38−5q−40 + 4q−42 + q−44−q−46 + 5q−48 + 6q−50−q−52−2q−54 + 2q−56−3q−58−8q−60−2q−62 + q−64−4q−66−2q−68 + 2q−70 + 2q−72 + q−78 + q−80
1,0,0,0 q−2 + 2q−6 + q−8 + q−10 + 2q−12−q−14−3q−18−q−20−2q−22 + q−26 + 3q−28 + 3q−30 + 2q−32 + 3q−34−q−36−2q−40−2q−42−q−44−q−46

B2 Invariants.

Weight Invariant
0,1 1−q−2 + 4q−4−4q−6 + 6q−8−6q−10 + 7q−12−6q−14 + 4q−16−3q−18−q−20 + 4q−22−8q−24 + 11q−26−11q−28 + 13q−30−11q−32 + 11q−34−7q−36 + 5q−38−q−40−2q−42 + 4q−44−6q−46 + 6q−48−7q−50 + 5q−52−4q−54 + 3q−56−2q−58 + q−60−q−62
1,0 q2−q−2−q−4 + 3q−6 + 3q−8−2q−10−4q−12 + q−14 + 6q−16 + 3q−18−6q−20−5q−22 + 4q−24 + 7q−26−7q−30−3q−32 + 5q−34 + 5q−36−2q−38−4q−40 + 2q−42 + 4q−44−4q−48 + 5q−52 + 2q−54−5q−56−3q−58 + 4q−60 + 5q−62−3q−64−6q−66 + 6q−70 + q−72−6q−74−5q−76 + 2q−78 + 6q−80−4q−84−3q−86 + q−88 + 3q−90 + q−92−q−94−q−96 + q−100

D4 Invariants.

Weight Invariant
1,0,0,0 q−2−q−4 + 3q−6−2q−8 + 6q−10−3q−12 + 6q−14−4q−16 + 7q−18−5q−20 + 2q−22−4q−24−q−28−4q−30 + 4q−32−6q−34 + 9q−36−7q−38 + 12q−40−6q−42 + 12q−44−7q−46 + 9q−48−5q−50 + 4q−52−4q−54−2q−56−q−58−4q−60 + 3q−62−6q−64 + 3q−66−5q−68 + 6q−70−4q−72 + 2q−74−3q−76 + 3q−78−q−80 + q−82−q−84 + q−86

G2 Invariants.

Weight Invariant
1,0 q−2−q−4 + 4q−6−5q−8 + 5q−10−2q−12−4q−14 + 14q−16−17q−18 + 19q−20−11q−22−2q−24 + 18q−26−27q−28 + 28q−30−17q−32 + q−34 + 13q−36−23q−38 + 20q−40−9q−42−5q−44 + 15q−46−18q−48 + 9q−50 + 3q−52−18q−54 + 24q−56−24q−58 + 16q−60 + q−62−18q−64 + 32q−66−33q−68 + 28q−70−9q−72−10q−74 + 25q−76−28q−78 + 24q−80−7q−82−7q−84 + 19q−86−16q−88 + 6q−90 + 6q−92−16q−94 + 18q−96−11q−98−2q−100 + 11q−102−18q−104 + 20q−106−15q−108 + 4q−110 + 3q−112−12q−114 + 12q−116−13q−118 + 10q−120−5q−122 + q−124 + 3q−126−7q−128 + 6q−130−5q−132 + 4q−134−2q−136 + q−140−2q−142 + 2q−144−q−146 + q−148

.

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 36"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t3 + 5t2−8t + 9−8t−1 + 5t−2−t−3
In[5]:= Conway[K][z]
Out[5]= −z6−z4 + 3z2 + 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]= { 37, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q9 + 2q8−4q7 + 6q6−6q5 + 6q4−5q3 + 4q2−2q + 1
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z6a−4 + z4a−2−4z4a−4 + 2z4a−6 + 3z2a−2−5z2a−4 + 6z2a−6−z2a−8 + 2a−2−3a−4 + 4a−6−2a−8
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z8a−4 + z8a−6 + 2z7a−3 + 5z7a−5 + 3z7a−7 + z6a−2 + z6a−4 + 4z6a−6 + 4z6a−8−7z5a−3−14z5a−5−4z5a−7 + 3z5a−9−4z4a−2−12z4a−4−17z4a−6−7z4a−8 + 2z4a−10 + 6z3a−3 + 9z3a−5−2z3a−9 + z3a−11 + 5z2a−2 + 12z2a−4 + 15z2a−6 + 7z2a−8−z2a−10−za−3−2za−5 + za−7 + za−9−za−11−2a−2−3a−4−4a−6−2a−8

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n16,}

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 36"];
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]= { −t3 + 5t2−8t + 9−8t−1 + 5t−2−t−3, −q9 + 2q8−4q7 + 6q6−6q5 + 6q4−5q3 + 4q2−2q + 1 }
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]= {K11n16,}

[edit] Vassiliev invariants

V2 and V3: (3, 7)
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
12 56 72 270 42 672 \frac{4304}{3} \frac{704}{3} 216 288 1568 3240 504 \frac{78751}{10} \frac{138}{5} \frac{48742}{15} \frac{577}{6} \frac{4191}{10}

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 = 4 is the signature of 9 36. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-101234567χ
19         1-1
17        1 1
15       31 -2
13      31  2
11     33   0
9    33    0
7   23     1
5  23      -1
3 13       2
1 1        -1
-11         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 3 i = 5
r = −2 {\mathbb Z}
r = −1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r = 1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 7 {\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 q25−2q24 + q23 + 3q22−8q21 + 6q20 + 7q19−20q18 + 13q17 + 14q16−32q15 + 15q14 + 21q13−35q12 + 10q11 + 25q10−30q9 + 2q8 + 24q7−19q6−4q5 + 17q4−8q3−5q2 + 7q−1−2q−1 + q−2
3 −q48 + 2q47−q46−q44 + 3q43−3q42−q41 + 5q40 + 2q39−14q38 + q37 + 23q36 + 2q35−40q34−5q33 + 57q32 + 10q31−67q30−24q29 + 79q28 + 32q27−77q26−46q25 + 75q24 + 51q23−62q22−61q21 + 51q20 + 63q19−34q18−68q17 + 22q16 + 64q15−3q14−63q13−8q12 + 53q11 + 22q10−44q9−26q8 + 27q7 + 32q6−17q5−25q4 + 4q3 + 20q2 + q−11−4q−1 + 6q−2 + 2q−3−q−4−2q−5 + q−6
4 q78−2q77 + q76−2q74 + 6q73−6q72 + 3q71−2q70−7q69 + 19q68−10q67 + 4q66−14q65−21q64 + 52q63 + 5q62 + 3q61−61q60−66q59 + 111q58 + 68q57 + 29q56−144q55−177q54 + 155q53 + 175q52 + 117q51−210q50−326q49 + 139q48 + 251q47 + 235q46−200q45−431q44 + 76q43 + 245q42 + 314q41−135q40−445q39 + 20q38 + 175q37 + 329q36−56q35−393q34−18q33 + 84q32 + 306q31 + 21q30−308q29−51q28−16q27 + 263q26 + 97q25−201q24−73q23−113q22 + 186q21 + 146q20−75q19−51q18−177q17 + 75q16 + 134q15 + 27q14 + 14q13−164q12−21q11 + 61q10 + 56q9 + 71q8−89q7−48q6−7q5 + 24q4 + 69q3−20q2−22q−23−6q−1 + 32q−2 + 2q−3−9q−5−8q−6 + 7q−7 + q−8 + 2q−9−q−10−2q−11 + q−12
5 −q115 + 2q114−q113 + 2q111−3q110−3q109 + 6q108−2q106 + 4q105−8q104−7q103 + 15q102 + 9q101−4q100−9q99−26q98−10q97 + 41q96 + 56q95 + 8q94−63q93−114q92−46q91 + 120q90 + 211q89 + 105q88−164q87−365q86−227q85 + 212q84 + 551q83 + 407q82−198q81−767q80−664q79 + 141q78 + 957q77 + 954q76−5q75−1088q74−1248q73−203q72 + 1160q71 + 1499q70 + 411q69−1123q68−1663q67−648q66 + 1043q65 + 1761q64 + 796q63−906q62−1745q61−934q60 + 770q59 + 1701q58 + 976q57−635q56−1584q55−1012q54 + 508q53 + 1479q52 + 997q51−383q50−1336q49−1005q48 + 256q47 + 1206q46 + 984q45−108q44−1037q43−988q42−49q41 + 869q40 + 941q39 + 217q38−642q37−900q36−367q35 + 421q34 + 777q33 + 489q32−167q31−634q30−548q29−45q28 + 420q27 + 541q26 + 231q25−218q24−447q23−327q22−q21 + 312q20 + 359q19 + 134q18−141q17−288q16−236q15−q14 + 199q13 + 225q12 + 103q11−69q10−188q9−145q8−10q7 + 102q6 + 134q5 + 67q4−38q3−91q2−74q−13 + 49q−1 + 61q−2 + 23q−3−11q−4−33q−5−30q−6−2q−7 + 19q−8 + 13q−9 + 6q−10−11q−12−6q−13 + 3q−14 + 2q−15 + q−16 + 2q−17−q−18−2q−19 + q−20
6 q159−2q158 + q157−2q155 + 3q154 + 3q152−9q151 + 4q150 + 5q149−9q148 + 7q147 + q146 + 2q145−22q144 + 14q143 + 23q142−17q141 + 7q140−8q139−20q138−50q137 + 49q136 + 91q135 + 4q134 + 7q133−74q132−140q131−147q130 + 126q129 + 324q128 + 197q127 + 80q126−272q125−576q124−535q123 + 171q122 + 880q121 + 896q120 + 532q119−526q118−1563q117−1660q116−215q115 + 1649q114 + 2360q113 + 1889q112−320q111−2866q110−3714q109−1619q108 + 1932q107 + 4143q106 + 4224q105 + 980q104−3574q103−5994q102−3962q101 + 1053q100 + 5152q99 + 6605q98 + 3142q97−3008q96−7301q95−6185q94−650q93 + 4821q92 + 7869q91 + 5049q90−1629q89−7217q88−7257q87−2117q86 + 3715q85 + 7801q84 + 5887q83−398q82−6372q81−7173q80−2778q79 + 2663q78 + 7061q77 + 5837q76 + 329q75−5445q74−6593q73−2950q72 + 1847q71 + 6211q70 + 5538q69 + 906q68−4520q67−5963q66−3170q65 + 929q64 + 5266q63 + 5298q62 + 1723q61−3306q60−5211q59−3559q58−342q57 + 3938q56 + 4911q55 + 2725q54−1636q53−3997q52−3749q51−1783q50 + 2085q49 + 3939q48 + 3405q47 + 222q46−2163q45−3196q44−2783q43 + 36q42 + 2210q41 + 3149q40 + 1556q39−113q38−1741q37−2679q36−1399q35 + 225q34 + 1847q33 + 1679q32 + 1255q31−4q30−1467q29−1540q28−1026q27 + 263q26 + 705q25 + 1311q24 + 974q23−55q22−651q21−1006q20−551q19−358q18 + 482q17 + 801q16 + 553q15 + 208q14−284q13−370q12−649q11−190q10 + 160q9 + 328q8 + 364q7 + 185q6 + 75q5−326q4−247q3−156q2−12q + 110 + 165q−1 + 202q−2−36q−3−56q−4−104q−5−80q−6−44q−7 + 24q−8 + 103q−9 + 22q−10 + 24q−11−13q−12−24q−13−39q−14−16q−15 + 24q−16 + 3q−17 + 14q−18 + 5q−19 + 3q−20−11q−21−8q−22 + 5q−23−2q−24 + 2q−25 + q−26 + 2q−27−q−28−2q−29 + q−30
7 −q210 + 2q209−q208 + 2q206−3q205 + 5q201−7q200 + 10q198−7q197−2q195 + 10q193−23q192−2q191 + 30q190 + 2q189 + 8q188−13q187−16q186−2q185−59q184−4q183 + 80q182 + 66q181 + 72q180−25q179−111q178−128q177−195q176−22q175 + 234q174 + 355q173 + 396q172 + 55q171−391q170−691q169−824q168−287q167 + 598q166 + 1339q165 + 1627q164 + 762q163−796q162−2263q161−2972q160−1774q159 + 793q158 + 3493q157 + 5011q156 + 3568q155−291q154−4852q153−7804q152−6380q151−985q150 + 5979q149 + 11075q148 + 10283q147 + 3487q146−6418q145−14563q144−15080q143−7200q142 + 5766q141 + 17506q140 + 20258q139 + 12093q138−3700q137−19423q136−25223q135−17618q134 + 357q133 + 19939q132 + 29168q131 + 23038q130 + 3952q129−18853q128−31698q127−27838q126−8519q125 + 16685q124 + 32629q123 + 31204q122 + 12699q121−13680q120−32121q119−33229q118−16054q117 + 10735q116 + 30705q115 + 33766q114 + 18191q113−8068q112−28724q111−33353q110−19364q109 + 6072q108 + 26814q107 + 32294q106 + 19632q105−4667q104−24989q103−31027q102−19553q101 + 3673q100 + 23484q99 + 29786q98 + 19309q97−2816q96−22038q95−28674q94−19251q93 + 1810q92 + 20571q91 + 27645q90 + 19425q89−448q88−18811q87−26599q86−19868q85−1347q84 + 16644q83 + 25314q82 + 20431q81 + 3602q80−13898q79−23673q78−20966q77−6153q76 + 10627q75 + 21432q74 + 21134q73 + 8861q72−6795q71−18529q70−20805q69−11345q68 + 2718q67 + 14866q66 + 19571q65 + 13324q64 + 1475q63−10618q62−17429q61−14381q60−5224q59 + 5978q58 + 14230q57 + 14294q56 + 8187q55−1397q54−10277q53−12873q52−9924q51−2639q50 + 5923q49 + 10292q48 + 10201q47 + 5568q46−1731q45−6854q44−9031q43−7126q42−1705q41 + 3248q40 + 6739q39 + 7095q38 + 3871q37 + 47q36−3839q35−5856q34−4688q33−2293q32 + 1104q31 + 3757q30 + 4129q29 + 3389q28 + 1072q27−1580q26−2836q25−3283q24−2149q23−197q22 + 1173q21 + 2391q20 + 2332q19 + 1230q18 + 150q17−1205q16−1771q15−1450q14−957q13 + 153q12 + 942q11 + 1134q10 + 1164q9 + 478q8−209q7−592q6−925q5−656q4−253q3 + 75q2 + 539q + 567 + 390q−1 + 192q−2−194q−3−300q−4−314q−5−301q−6−38q−7 + 117q−8 + 195q−9 + 232q−10 + 85q−11 + 17q−12−43q−13−148q−14−102q−15−57q−16 + 4q−17 + 74q−18 + 41q−19 + 40q−20 + 37q−21−15q−22−27q−23−34q−24−22q−25 + 13q−26 + q−27 + 5q−28 + 16q−29 + 5q−30 + 2q−31−8q−32−8q−33 + 3q−34−2q−36 + 2q−37 + q−38 + 2q−39−q−40−2q−41 + q−42

[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_35

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