8 7

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8_6

8_8

Contents

Image:8 7.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8_7's page at Knotilus!

Visit 8 7's page at the original Knot Atlas!

[edit 8 7 Quick Notes]
[edit 8 7 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X7,15,8,14 X13,7,14,6 X15,9,16,8 X9,2,10,3
Gauss code -1, 8, -2, 1, -4, 6, -5, 7, -8, 2, -3, 4, -6, 5, -7, 3
Dowker-Thistlethwaite code 4 10 12 14 2 16 6 8
Conway Notation [4112]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 7]

Knot 8_7.
Knot 8_7.
A graph, knot 8_7.
A graph, knot 8_7.
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["8 7"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X7,15,8,14 X13,7,14,6 X15,9,16,8 X9,2,10,3
In[5]:= GaussCode[K]
Out[5]= -1, 8, -2, 1, -4, 6, -5, 7, -8, 2, -3, 4, -6, 5, -7, 3
In[6]:= DTCode[K]
Out[6]= 4 10 12 14 2 16 6 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [4112]
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,−2,1,−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, 8, 3 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.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:8 7_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{10, 6}, {1, 8}, {7, 9}, {8, 10}, {9, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 1}, {2, 7}]
In[14]:= Draw[ap]
Image:8 7_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index {3,6}
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-8]
Hyperbolic Volume 7.0222
A-Polynomial See Data:8 7/A-polynomial

[edit Notes for 8 7's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for 8 7's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t3−3t2 + 5t−5 + 5t−1−3t−2 + t−3
Conway polynomial z6 + 3z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 23, 2 }
Jones polynomial −q6 + 2q5−3q4 + 4q3−4q2 + 4q−2 + 2q−1−q−2
HOMFLY-PT polynomial (db, data sources) z6a−2 + 5z4a−2−z4a−4−z4 + 8z2a−2−3z2a−4−3z2 + 4a−2−2a−4−1
Kauffman polynomial (db, data sources) z7a−1 + z7a−3 + 4z6a−2 + 2z6a−4 + 2z6 + az5−z5a−1 + 2z5a−5−12z4a−2−3z4a−4 + 2z4a−6−7z4−3az3−3z3a−1−2z3a−3−z3a−5 + z3a−7 + 12z2a−2 + 4z2a−4−2z2a−6 + 6z2 + az + 2za−1 + 2za−3−za−7−4a−2−2a−4−1
The A2 invariant −q6 + 1 + 2q−2 + 2q−6 + q−10−q−14−q−18
The G2 invariant q32−q30 + 2q28−3q26 + q24−q22−3q20 + 7q18−8q16 + 5q14−3q12−2q10 + 6q8−10q6 + 7q4−3q2 + 6q−2−6q−4 + 4q−6 + 3q−8−q−10 + 4q−12−4q−14 + 2q−16 + 4q−18−5q−20 + 10q−22−7q−24 + 5q−26 + 3q−28−7q−30 + 10q−32−11q−34 + 8q−36−2q−38−3q−40 + 8q−42−8q−44 + 6q−46−3q−50 + 3q−52−3q−54−q−56 + 4q−58−5q−60 + 5q−62−3q−64−2q−66 + 4q−68−7q−70 + 5q−72−5q−74 + 2q−76−q−78−3q−80 + 4q−82−4q−84 + 4q−86−2q−88 + q−90−2q−94 + 2q−96−q−98 + q−100
Further Quantum Invariants
Further quantum knot invariants for 8_7.

A1 Invariants.

Weight Invariant
1 −q5 + q3 + 2q−1 + q−7−q−9 + q−11−q−13
2 q16−q14−2q12 + 2q10−3q6 + 2q4 + 3q2−2 + 2q−2 + 3q−4−2q−6 + 2q−10−2q−14−q−16 + 3q−18−2q−20−2q−22 + 4q−24−q−26−2q−28 + 2q−30−q−32−q−34 + q−36
3 −q33 + q31 + 2q29−3q25−2q23 + 4q21 + 3q19−4q17−6q15 + q13 + 6q11 + 2q9−7q7−2q5 + 6q3 + 7q−3q−1−5q−3 + 3q−5 + 7q−7−q−9−6q−11 + 2q−13 + 5q−15−6q−19−2q−21 + 3q−23 + 3q−25−q−27−5q−29−2q−31 + 7q−33 + 4q−35−6q−37−6q−39 + 5q−41 + 6q−43−3q−45−5q−47 + 2q−49 + 3q−51−q−53−2q−55 + q−57−q−61 + q−65 + q−67−q−69
4 q56−q54−2q52 + q48 + 5q46−4q42−4q40−3q38 + 10q36 + 7q34−2q32−8q30−13q28 + 4q26 + 10q24 + 9q22−18q18−8q16 + 2q14 + 15q12 + 14q10−8q8−13q6−12q4 + 10q2 + 23 + 5q−2−9q−4−20q−6 + 21q−10 + 10q−12−5q−14−19q−16−4q−18 + 15q−20 + 8q−22−4q−24−15q−26−4q−28 + 10q−30 + 10q−32−2q−34−10q−36−6q−38 + q−40 + 11q−42 + 7q−44 + q−46−13q−48−15q−50 + 8q−52 + 15q−54 + 14q−56−10q−58−25q−60−3q−62 + 12q−64 + 23q−66 + q−68−21q−70−9q−72 + 16q−76 + 7q−78−8q−80−4q−82−6q−84 + 5q−86 + 4q−88−q−90 + 2q−92−5q−94 + q−98 + 3q−102−q−104−q−108−q−110 + q−112
5 −q85 + q83 + 2q81−q77−3q75−3q73 + 6q69 + 6q67 + q65−5q63−10q61−8q59 + 3q57 + 16q55 + 15q53 + 3q51−11q49−22q47−15q45 + 4q43 + 22q41 + 23q39 + 9q37−12q35−30q33−24q31−3q29 + 22q27 + 33q25 + 21q23−9q21−34q19−38q17−14q15 + 28q13 + 51q11 + 31q9−8q7−48q5−50q3−5q + 48q−1 + 59q−3 + 23q−5−33q−7−63q−9−34q−11 + 25q−13 + 60q−15 + 39q−17−15q−19−52q−21−40q−23 + 7q−25 + 45q−27 + 34q−29−7q−31−36q−33−30q−35 + 4q−37 + 33q−39 + 24q−41−7q−43−26q−45−20q−47 + 2q−49 + 23q−51 + 20q−53 + 5q−55−15q−57−23q−59−15q−61 + 3q−63 + 25q−65 + 32q−67 + 14q−69−24q−71−47q−73−31q−75 + 12q−77 + 54q−79 + 54q−81 + 2q−83−57q−85−69q−87−21q−89 + 46q−91 + 77q−93 + 39q−95−30q−97−71q−99−49q−101 + 13q−103 + 56q−105 + 49q−107 + 3q−109−37q−111−43q−113−12q−115 + 22q−117 + 29q−119 + 14q−121−7q−123−20q−125−14q−127 + 2q−129 + 11q−131 + 10q−133 + 3q−135−5q−137−8q−139−4q−141 + 3q−143 + 5q−145 + 2q−147 + q−149−2q−151−3q−153−q−155 + q−157 + q−161 + q−163−q−165
6 q120−q118−2q116 + q112 + 3q110 + q108 + 3q106−2q104−8q102−5q100−q98 + 6q96 + 7q94 + 14q92 + 3q90−12q88−18q86−17q84−3q82 + 6q80 + 33q78 + 30q76 + 8q74−15q72−34q70−34q68−29q66 + 17q64 + 42q62 + 47q60 + 28q58−q56−31q54−67q52−42q50−12q48 + 30q46 + 56q44 + 66q42 + 42q40−27q38−62q36−86q34−62q32−8q30 + 75q28 + 117q26 + 81q24 + 18q22−81q20−134q18−123q16−12q14 + 104q12 + 154q10 + 136q8 + 15q6−116q4−190q2−123 + 12q−2 + 137q−4 + 195q−6 + 116q−8−37q−10−177q−12−176q−14−74q−16 + 70q−18 + 179q−20 + 159q−22 + 32q−24−123q−26−164q−28−104q−30 + 17q−32 + 128q−34 + 140q−36 + 53q−38−75q−40−117q−42−85q−44 + 2q−46 + 83q−48 + 94q−50 + 35q−52−51q−54−74q−56−49q−58 + 13q−60 + 56q−62 + 58q−64 + 18q−66−36q−68−54q−70−42q−72 + 5q−74 + 37q−76 + 54q−78 + 40q−80 + 2q−82−46q−84−73q−86−52q−88−13q−90 + 59q−92 + 102q−94 + 94q−96 + 4q−98−96q−100−138q−102−118q−104 + 5q−106 + 136q−108 + 202q−110 + 116q−112−44q−114−177q−116−226q−118−111q−120 + 80q−122 + 236q−124 + 213q−126 + 65q−128−113q−130−237q−132−188q−134−24q−136 + 157q−138 + 200q−140 + 125q−142−8q−144−144q−146−162q−148−76q−150 + 53q−152 + 108q−154 + 97q−156 + 40q−158−50q−160−84q−162−59q−164 + 4q−166 + 35q−168 + 45q−170 + 37q−172−10q−174−33q−176−29q−178−4q−180 + 7q−182 + 16q−184 + 23q−186−12q−190−14q−192−4q−194−q−196 + 5q−198 + 13q−200 + 2q−202−3q−204−6q−206−2q−208−3q−210 + 5q−214 + q−216 + q−218−q−220−q−224−q−226 + q−228

A2 Invariants.

Weight Invariant
1,0 −q6 + 1 + 2q−2 + 2q−6 + q−10−q−14−q−18
1,1 q20−2q18 + 4q16−8q14 + 13q12−18q10 + 18q8−24q6 + 20q4−16q2 + 10 + 4q−2−4q−4 + 22q−6−22q−8 + 34q−10−34q−12 + 34q−14−34q−16 + 26q−18−22q−20 + 12q−22−4q−24−4q−26 + 7q−28−12q−30 + 14q−32−14q−34 + 13q−36−12q−38 + 12q−40−10q−42 + 7q−44−6q−46 + 4q−48−2q−50 + q−52
2,0 q18−q14−q12−q8−3q6 + 2q2 + 2 + q−2 + 4q−4 + 2q−6 + q−8 + 2q−10 + 2q−12−q−16 + q−18−q−20−3q−22 + q−26−q−28−q−30 + q−32−q−36−q−38 + q−46
3,0 −q36 + q32 + 2q30 + q28−2q26−q24 + q22 + 4q20−6q16−7q14−q12 + 3q10−6q6−4q4 + 5q2 + 10 + 6q−2−2q−4 + q−6 + 5q−8 + 8q−10 + q−12 + 2q−16 + 4q−18 + q−20−4q−22−q−24−3q−28−7q−30−5q−32 + 3q−34 + 3q−36−3q−38−6q−40 + q−42 + 8q−44 + 5q−46−3q−48−6q−50 + q−52 + 6q−54 + 2q−56−4q−58−5q−60 + 3q−64−q−68−2q−70 + q−72 + q−74 + q−76 + q−78−q−84

A3 Invariants.

Weight Invariant
0,1,0 q14−q12−3q6 + q4−2 + 2q−2 + 3q−4−q−6 + 4q−8 + 5q−10 + 2q−12 + 2q−14 + q−16−3q−20−3q−22 + q−24−3q−26−2q−28 + 3q−30−q−32−2q−34 + 2q−36−q−40 + q−42
1,0,0 −q7−q3 + q + 2q−3 + q−5 + 2q−7 + 2q−9 + q−11 + q−13−q−15−2q−19−q−23
1,0,1 q24−2q22 + 3q20−3q18 + q16 + 5q14−10q12 + 11q10−12q8 + q6 + q4−16q2 + 16−18q−2 + 15q−4 + 6q−6 + 22q−10 + 2q−12 + 11q−14 + 4q−16 + 4q−18−8q−20 + 7q−22−16q−24−18q−30 + 16q−32−15q−34 + 2q−36 + 9q−38−13q−40 + 10q−42−q−44−6q−46 + 11q−48−7q−50 + 2q−52 + 4q−54−8q−56 + 6q−58−3q−60−q−62 + 3q−64−2q−66 + q−68

A4 Invariants.

Weight Invariant
0,1,0,0 q16−q8−2q6−q4−2q2−3 + q−4 + q−6 + q−8 + 7q−10 + 7q−12 + 5q−14 + 6q−16 + 8q−18 + q−20−q−22−4q−26−6q−28−3q−30−2q−32−4q−34−2q−36 + q−38−2q−42 + q−44 + 2q−46 + q−52
1,0,0,0 −q8−q4 + 2q−4 + q−6 + 3q−8 + 2q−10 + 3q−12 + q−14 + q−16−q−18−q−20−q−22−2q−24−q−28

B2 Invariants.

Weight Invariant
0,1 −q14 + q12−2q10 + 2q8−3q6 + 3q4−2q2 + 2 + q−4 + 3q−6−2q−8 + 5q−10−4q−12 + 6q−14−5q−16 + 4q−18−3q−20 + q−22−q−24−q−26 + 2q−28−3q−30 + 3q−32−2q−34 + 2q−36−2q−38 + q−40−q−42
1,0 q24−q20−q18 + q16 + q14−2q12−3q10 + 3q6 + q4−2q2−2 + 2q−2 + 3q−4 + 2q−6−2q−8 + 2q−12 + 3q−14 + 2q−20 + 3q−22−2q−26 + 2q−30−3q−34−2q−36 + q−38 + q−40−2q−42−3q−44 + 3q−48 + q−50−2q−52−2q−54 + 2q−58 + q−60−q−62−q−64 + q−68

D4 Invariants.

Weight Invariant
1,0,0,0 q18−q16 + q14−2q12 + 2q10−3q8 + q6−3q4 + q2−2 + q−6 + 3q−8 + q−10 + 7q−12 + 2q−14 + 8q−16−q−18 + 6q−20−3q−22 + 3q−24−5q−26−q−28−4q−30−q−32−q−34−2q−36−2q−40 + 3q−42−2q−44 + q−46−2q−48 + 2q−50−q−52 + q−54−q−56 + q−58

G2 Invariants.

Weight Invariant
1,0 q32−q30 + 2q28−3q26 + q24−q22−3q20 + 7q18−8q16 + 5q14−3q12−2q10 + 6q8−10q6 + 7q4−3q2 + 6q−2−6q−4 + 4q−6 + 3q−8−q−10 + 4q−12−4q−14 + 2q−16 + 4q−18−5q−20 + 10q−22−7q−24 + 5q−26 + 3q−28−7q−30 + 10q−32−11q−34 + 8q−36−2q−38−3q−40 + 8q−42−8q−44 + 6q−46−3q−50 + 3q−52−3q−54−q−56 + 4q−58−5q−60 + 5q−62−3q−64−2q−66 + 4q−68−7q−70 + 5q−72−5q−74 + 2q−76−q−78−3q−80 + 4q−82−4q−84 + 4q−86−2q−88 + q−90−2q−94 + 2q−96−q−98 + q−100

.

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["8 7"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t3−3t2 + 5t−5 + 5t−1−3t−2 + t−3
In[5]:= Conway[K][z]
Out[5]= z6 + 3z4 + 2z2 + 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]= { 23, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q6 + 2q5−3q4 + 4q3−4q2 + 4q−2 + 2q−1−q−2
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z6a−2 + 5z4a−2−z4a−4−z4 + 8z2a−2−3z2a−4−3z2 + 4a−2−2a−4−1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z7a−1 + z7a−3 + 4z6a−2 + 2z6a−4 + 2z6 + az5−z5a−1 + 2z5a−5−12z4a−2−3z4a−4 + 2z4a−6−7z4−3az3−3z3a−1−2z3a−3−z3a−5 + z3a−7 + 12z2a−2 + 4z2a−4−2z2a−6 + 6z2 + az + 2za−1 + 2za−3−za−7−4a−2−2a−4−1

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

Same Alexander/Conway Polynomial: {K11n24,}

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["8 7"];
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]= { t3−3t2 + 5t−5 + 5t−1−3t−2 + t−3, −q6 + 2q5−3q4 + 4q3−4q2 + 4q−2 + 2q−1−q−2 }
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]= {K11n24,}
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: (2, 2)

[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 = 2 is the signature of 8 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-1012345χ
13        1-1
11       1 1
9      21 -1
7     21  1
5    22   0
3   22    0
1  13     2
-1 11      0
-3 1       1
-51        -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −3 {\mathbb Z}
r = −2 {\mathbb Z}\oplus{\mathbb Z}_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}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 5 {\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 q17−2q16 + 4q14−6q13 + q12 + 9q11−12q10 + q9 + 14q8−16q7 + 16q5−14q4−2q3 + 14q2−9q−3 + 10q−1−4q−2−4q−3 + 5q−4−q−5−2q−6 + q−7
3 −q33 + 2q32−q30−2q29 + 3q28 + q27−4q26−q25 + 7q24−11q22 + q21 + 16q20−q19−22q18 + q17 + 26q16 + 2q15−31q14−2q13 + 30q12 + 6q11−31q10−7q9 + 26q8 + 12q7−26q6−10q5 + 18q4 + 17q3−18q2−14q + 10 + 19q−1−8q−2−15q−3 + 2q−4 + 14q−5 + q−6−11q−7−3q−8 + 7q−9 + 3q−10−4q−11−2q−12 + q−13 + 2q−14−q−15
4 q54−2q53 + q51−q50 + 5q49−5q48 + q47−6q45 + 12q44−8q43 + 6q42 + q41−17q40 + 14q39−12q38 + 21q37 + 10q36−33q35 + 5q34−24q33 + 43q32 + 32q31−44q30−10q29−46q28 + 58q27 + 56q26−43q25−17q24−69q23 + 60q22 + 70q21−37q20−13q19−79q18 + 53q17 + 66q16−29q15−q14−79q13 + 39q12 + 55q11−18q10 + 11q9−72q8 + 20q7 + 40q6−4q5 + 26q4−61q3−q2 + 20q + 7 + 40q−1−43q−2−14q−3−2q−4 + 6q−5 + 45q−6−21q−7−13q−8−15q−9−4q−10 + 35q−11−3q−12−4q−13−14q−14−10q−15 + 18q−16 + 2q−17 + 2q−18−5q−19−7q−20 + 5q−21 + q−22 + 2q−23−q−24−2q−25 + q−26
5 −q80 + 2q79−q77 + q76−2q75−3q74 + 3q73 + 3q72 + 4q70−4q69−10q68−q67 + 6q66 + 8q65 + 11q64−3q63−19q62−17q61 + 21q59 + 32q58 + 12q57−26q56−51q55−31q54 + 27q53 + 72q52 + 58q51−19q50−94q49−93q48 + 5q47 + 113q46 + 127q45 + 19q44−125q43−160q42−43q41 + 125q40 + 186q39 + 71q38−125q37−202q36−86q35 + 109q34 + 209q33 + 109q32−107q31−209q30−108q29 + 91q28 + 201q27 + 117q26−87q25−194q24−105q23 + 70q22 + 179q21 + 111q20−68q19−163q18−96q17 + 41q16 + 145q15 + 105q14−39q13−122q12−85q11 + 3q10 + 98q9 + 93q8−2q7−68q6−64q5−32q4 + 39q3 + 64q2 + 28q−12−28q−1−43q−2−14q−3 + 19q−4 + 30q−5 + 28q−6 + 11q−7−23q−8−37q−9−23q−10 + 6q−11 + 32q−12 + 36q−13 + 7q−14−25q−15−34q−16−19q−17 + 11q−18 + 30q−19 + 25q−20−4q−21−20q−22−20q−23−7q−24 + 11q−25 + 18q−26 + 7q−27−6q−28−8q−29−6q−30−2q−31 + 7q−32 + 5q−33−q−34−2q−35−q−36−2q−37 + q−38 + 2q−39−q−40
6 q111−2q110 + q108−q107 + 2q106 + 5q104−7q103−3q102 + 2q101−5q100 + 5q99 + 5q98 + 16q97−15q96−9q95−q94−15q93 + 7q92 + 17q91 + 39q90−22q89−18q88−12q87−40q86 + 3q85 + 40q84 + 86q83−14q82−28q81−43q80−103q79−22q78 + 74q77 + 176q76 + 43q75−17q74−98q73−232q72−108q71 + 92q70 + 312q69 + 176q68 + 58q67−141q66−413q65−272q64 + 43q63 + 436q62 + 354q61 + 206q60−118q59−569q58−463q57−72q56 + 485q55 + 487q54 + 366q53−32q52−635q51−594q50−195q49 + 465q48 + 529q47 + 466q46 + 58q45−627q44−637q43−267q42 + 426q41 + 508q40 + 493q39 + 106q38−589q37−623q36−284q35 + 394q34 + 461q33 + 481q32 + 124q31−535q30−583q29−286q28 + 351q27 + 399q26 + 456q25 + 147q24−449q23−524q22−297q21 + 270q20 + 312q19 + 424q18 + 189q17−321q16−437q15−309q14 + 159q13 + 191q12 + 364q11 + 230q10−166q9−310q8−289q7 + 50q6 + 47q5 + 262q4 + 229q3−26q2−157q−210−8q−1−78q−2 + 127q−3 + 162q−4 + 48q−5−26q−6−89q−7 + 10q−8−128q−9 + 11q−10 + 51q−11 + 37q−12 + 27q−13 + 10q−14 + 73q−15−92q−16−31q−17−32q−18−17q−19 + 3q−20 + 34q−21 + 108q−22−23q−23−7q−24−42q−25−43q−26−39q−27 + 4q−28 + 83q−29 + 13q−30 + 23q−31−13q−32−24q−33−44q−34−21q−35 + 37q−36 + 8q−37 + 23q−38 + 6q−39−q−40−22q−41−18q−42 + 10q−43−q−44 + 9q−45 + 5q−46 + 5q−47−7q−48−7q−49 + 3q−50−2q−51 + 2q−52 + q−53 + 2q−54−q−55−2q−56 + q−57
7 −q147 + 2q146−q144 + q143−2q142−2q140−q139 + 7q138 + q137−q136 + 4q135−7q134−3q133−7q132−6q131 + 18q130 + 6q129 + 2q128 + 9q127−13q126−8q125−18q124−17q123 + 32q122 + 16q121 + 11q120 + 19q119−26q118−19q117−35q116−33q115 + 50q114 + 42q113 + 44q112 + 36q111−59q110−64q109−82q108−65q107 + 84q106 + 117q105 + 145q104 + 100q103−110q102−182q101−239q100−177q99 + 112q98 + 273q97 + 387q96 + 302q95−98q94−371q93−566q92−481q91 + 35q90 + 446q89 + 775q88 + 721q87 + 87q86−499q85−990q84−979q83−256q82 + 491q81 + 1163q80 + 1255q79 + 467q78−436q77−1292q76−1498q75−682q74 + 333q73 + 1361q72 + 1686q71 + 877q70−205q69−1364q68−1814q67−1047q66 + 81q65 + 1343q64 + 1884q63 + 1141q62 + 27q61−1276q60−1903q59−1218q58−113q57 + 1238q56 + 1897q55 + 1230q54 + 155q53−1174q52−1863q51−1238q50−193q49 + 1136q48 + 1838q47 + 1215q46 + 200q45−1082q44−1784q43−1203q42−233q41 + 1039q40 + 1741q39 + 1174q38 + 253q37−952q36−1669q35−1174q34−305q33 + 876q32 + 1589q31 + 1138q30 + 369q29−731q28−1485q27−1145q26−447q25 + 610q24 + 1354q23 + 1087q22 + 539q21−412q20−1199q19−1075q18−619q17 + 259q16 + 1017q15 + 966q14 + 687q13−38q12−808q11−897q10−728q9−101q8 + 599q7 + 719q6 + 716q5 + 273q4−369q3−582q2−668q−345 + 191q−1 + 370q−2 + 554q−3 + 398q−4−24q−5−204q−6−425q−7−363q−8−60q−9 + 41q−10 + 264q−11 + 294q−12 + 103q−13 + 60q−14−126q−15−191q−16−75q−17−113q−18 + 15q−19 + 86q−20 + 25q−21 + 109q−22 + 35q−23−q−24 + 48q−25−67q−26−60q−27−43q−28−89q−29 + 14q−30 + 23q−31 + 49q−32 + 128q−33 + 34q−34−34q−36−108q−37−52q−38−45q−39−4q−40 + 91q−41 + 66q−42 + 49q−43 + 20q−44−52q−45−37q−46−53q−47−45q−48 + 25q−49 + 29q−50 + 42q−51 + 33q−52−9q−53−5q−54−20q−55−33q−56−6q−57 + 2q−58 + 16q−59 + 19q−60−2q−61 + 4q−62−q−63−11q−64−5q−65−4q−66 + 4q−67 + 7q−68−q−69 + 2q−71−2q−72−q−73−2q−74 + q−75 + 2q−76−q−77

[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.

8_6

8_8

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