8 5

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8_6

Contents

Image:8 5.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8_5's page at Knotilus!

Visit 8 5's page at the original Knot Atlas!

[edit 8 5 Quick Notes]

8 5 is also known as the pretzel knot P(3,3,2).

[edit 8 5 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X8493 X2837 X14,10,15,9 X12,5,13,6 X4,13,5,14 X16,12,1,11 X10,16,11,15
Gauss code 1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 7, -5, 6, -4, 8, -7
Dowker-Thistlethwaite code 6 8 12 2 14 16 4 10
Conway Notation [3,3,2]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 5]

Knot 8_5.
Knot 8_5.
A graph, knot 8_5.
A graph, knot 8_5.
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 5"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X8493 X2837 X14,10,15,9 X12,5,13,6 X4,13,5,14 X16,12,1,11 X10,16,11,15
In[5]:= GaussCode[K]
Out[5]= 1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 7, -5, 6, -4, 8, -7
In[6]:= DTCode[K]
Out[6]= 6 8 12 2 14 16 4 10

(The path below may be different on your system)

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

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index {4,6}
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-11]
Hyperbolic Volume 6.99719
A-Polynomial See Data:8 5/A-polynomial

[edit Notes for 8 5'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 8 5's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t3 + 3t2−4t + 5−4t−1 + 3t−2−t−3
Conway polynomial −z6−3z4−z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 21, 4 }
Jones polynomial q8−2q7 + 3q6−4q5 + 3q4−3q3 + 3q2−q + 1
HOMFLY-PT polynomial (db, data sources) −z6a−4 + z4a−2−5z4a−4 + z4a−6 + 4z2a−2−8z2a−4 + 3z2a−6 + 4a−2−5a−4 + 2a−6
Kauffman polynomial (db, data sources) z7a−3 + z7a−5 + z6a−2 + 4z6a−4 + 3z6a−6−3z5a−3 + z5a−5 + 4z5a−7−5z4a−2−15z4a−4−7z4a−6 + 3z4a−8−10z3a−5−8z3a−7 + 2z3a−9 + 8z2a−2 + 15z2a−4 + 4z2a−6−2z2a−8 + z2a−10 + 3za−3 + 7za−5 + 4za−7−4a−2−5a−4−2a−6
The A2 invariant 1 + q−2 + 2q−4 + 2q−6−3q−12−q−14−q−16 + q−20 + q−24
The G2 invariant q−2 + 3q−6−2q−8 + 2q−10 + q−12−2q−14 + 7q−16−5q−18 + 5q−20 + q−22−2q−24 + 8q−26−5q−28 + 5q−30 + 2q−32−2q−34 + 3q−36−3q−38 + 2q−42−4q−44 + 2q−46−4q−48−2q−50 + 2q−52−9q−54 + 4q−56−7q−58 + q−62−7q−64 + 7q−66−7q−68 + 4q−70 + 2q−72−5q−74 + 5q−76−2q−78 + q−80 + 4q−82−2q−84 + 3q−86 + q−88−q−90 + 4q−92−4q−94 + 3q−96−2q−100 + 3q−102−3q−104 + 3q−106−q−108 + q−110−3q−114 + 2q−116−2q−118 + 2q−120−q−122−q−128 + q−130−q−132 + q−134
Further Quantum Invariants
Further quantum knot invariants for 8_5.

A1 Invariants.

Weight Invariant
1 q + 2q−3−q−9−q−11 + q−13−q−15 + q−17
2 q6−q2 + 2 + 2q−2−2q−4 + q−6 + 2q−8−2q−10−q−12 + 2q−14−q−16−2q−18 + q−20 + q−22−q−24 + 2q−28−2q−32 + 2q−34 + q−36−3q−38 + q−40−q−44 + q−46
3 q15−q11−q9 + 2q7 + 3q5−4q−q−1 + 4q−3 + 4q−5−3q−7−4q−9 + 5q−13 + 2q−15−4q−17−2q−19 + 3q−21 + 4q−23−2q−25−4q−27 + 5q−31−q−33−4q−35−q−37 + 6q−39 + q−41−5q−43−3q−45 + 3q−47 + 3q−49−q−51−3q−53−2q−55 + 4q−57 + 4q−59−2q−61−6q−63 + q−65 + 5q−67−3q−71 + q−73 + q−75−q−79−q−85 + q−87
4 q28−q24−q22−q20 + 3q18 + 3q16 + q14−2q12−7q10−q8 + 4q6 + 8q4 + 5q2−7−8q−2−6q−4 + 5q−6 + 13q−8 + 4q−10−3q−12−13q−14−7q−16 + 9q−18 + 12q−20 + 10q−22−7q−24−14q−26−4q−28 + 5q−30 + 15q−32 + 4q−34−11q−36−12q−38−2q−40 + 12q−42 + 8q−44−5q−46−12q−48−5q−50 + 10q−52 + 9q−54−4q−56−12q−58−3q−60 + 11q−62 + 10q−64−4q−66−12q−68−3q−70 + 8q−72 + 13q−74 + 2q−76−9q−78−9q−80−6q−82 + 7q−84 + 13q−86 + 5q−88−7q−90−19q−92−6q−94 + 14q−96 + 17q−98 + 5q−100−19q−102−15q−104 + 4q−106 + 14q−108 + 10q−110−8q−112−8q−114−q−116 + 3q−118 + 6q−120−3q−122−2q−124 + q−126 + 2q−130−2q−132−q−138 + q−140
5 q45−q41−q39−q37 + 3q33 + 4q31 + q29−2q27−5q25−7q23−2q21 + 6q19 + 11q17 + 9q15 + q13−11q11−16q9−11q7 + 3q5 + 17q3 + 21q + 10q−1−8q−3−23q−5−25q−7−6q−9 + 18q−11 + 30q−13 + 25q−15 + q−17−27q−19−35q−21−16q−23 + 14q−25 + 36q−27 + 36q−29 + 7q−31−28q−33−43q−35−26q−37 + 10q−39 + 41q−41 + 40q−43 + 6q−45−34q−47−46q−49−24q−51 + 17q−53 + 45q−55 + 33q−57−7q−59−38q−61−39q−63−5q−65 + 33q−67 + 39q−69 + 10q−71−23q−73−32q−75−11q−77 + 23q−79 + 29q−81 + 6q−83−18q−85−24q−87−5q−89 + 23q−91 + 24q−93−2q−95−25q−97−25q−99−2q−101 + 26q−103 + 29q−105 + 6q−107−23q−109−33q−111−18q−113 + 8q−115 + 30q−117 + 33q−119 + 11q−121−20q−123−40q−125−34q−127 + 42q−131 + 55q−133 + 23q−135−28q−137−63q−139−45q−141 + 12q−143 + 58q−145 + 57q−147 + 6q−149−47q−151−54q−153−19q−155 + 28q−157 + 45q−159 + 22q−161−17q−163−31q−165−16q−167 + 6q−169 + 19q−171 + 12q−173−4q−175−11q−177−4q−179 + 3q−181 + 4q−183 + q−185−q−187−2q−189 + 2q−193 + q−195−2q−197−q−203 + q−205
6 q66−q62−q60−q58 + 4q52 + 4q50 + q48−2q46−5q44−6q42−8q40 + q38 + 8q36 + 13q34 + 12q32 + 6q30−5q28−22q26−21q24−15q22 + 2q20 + 19q18 + 33q16 + 32q14 + 7q12−16q10−40q8−43q6−30q4 + 8q2 + 44 + 56q−2 + 50q−4 + 10q−6−34q−8−73q−10−67q−12−28q−14 + 23q−16 + 76q−18 + 88q−20 + 59q−22−13q−24−72q−26−98q−28−80q−30−11q−32 + 68q−34 + 117q−36 + 94q−38 + 29q−40−57q−42−123q−44−119q−46−44q−48 + 60q−50 + 121q−52 + 125q−54 + 56q−56−54q−58−137q−60−135q−62−52q−64 + 53q−66 + 136q−68 + 138q−70 + 53q−72−69q−74−141q−76−122q−78−36q−80 + 79q−82 + 144q−84 + 114q−86 + 8q−88−93q−90−129q−92−86q−94 + 16q−96 + 102q−98 + 114q−100 + 45q−102−42q−104−98q−106−85q−108−11q−110 + 63q−112 + 83q−114 + 36q−116−24q−118−64q−120−48q−122 + 5q−124 + 51q−126 + 52q−128−q−130−47q−132−60q−134−20q−136 + 37q−138 + 74q−140 + 58q−142−11q−144−68q−146−84q−148−42q−150 + 22q−152 + 81q−154 + 95q−156 + 49q−158−20q−160−83q−162−101q−164−75q−166−2q−168 + 85q−170 + 131q−172 + 106q−174 + 15q−176−90q−178−161q−180−143q−182−29q−184 + 114q−186 + 188q−188 + 151q−190 + 26q−192−127q−194−205q−196−149q−198 + q−200 + 138q−202 + 186q−204 + 124q−206−20q−208−142q−210−156q−212−67q−214 + 47q−216 + 114q−218 + 109q−220 + 26q−222−61q−224−86q−226−50q−228 + 7q−230 + 44q−232 + 52q−234 + 16q−236−24q−238−30q−240−14q−242 + 3q−244 + 11q−246 + 15q−248 + 3q−250−9q−252−5q−254 + 2q−258 + 3q−262−q−264−3q−266 + 2q−268 + q−270 + q−272−2q−274−q−280 + q−282

A2 Invariants.

Weight Invariant
1,0 1 + q−2 + 2q−4 + 2q−6−3q−12−q−14−q−16 + q−20 + q−24
1,1 q4 + 6−4q−2 + 12q−4−10q−6 + 16q−8−12q−10 + 10q−12−8q−14−2q−16 + 2q−18−14q−20 + 12q−22−22q−24 + 22q−26−20q−28 + 22q−30−12q−32 + 16q−34−q−36 + 4q−40−10q−42 + 5q−44−10q−46 + 6q−48−6q−50 + 5q−52−2q−54 + 4q−56−4q−58 + 3q−60−2q−62 + 2q−64−2q−66 + q−68
2,0 q4 + q2 + 1 + q−2 + 3q−4 + 3q−6 + 2q−8 + q−12−2q−14−4q−16−4q−18−3q−20−3q−22−q−24 + 2q−26 + 3q−28 + 3q−30 + 3q−32 + 3q−34−q−36 + q−40−2q−44−q−46−q−48−q−50−q−52 + q−56 + q−60
3,0 q12 + q10 + q8 + 2q2 + 4 + 4q−2 + q−4−q−6 + 3q−10 + 3q−12−2q−14−8q−16−6q−18−3q−20 + q−22−4q−24−5q−26−3q−28 + 5q−30 + 7q−32 + 6q−34 + 4q−36 + 6q−38 + 8q−40 + 5q−42 + q−44−3q−46−2q−48−6q−50−7q−52−8q−54−q−56−2q−58−2q−60−3q−62 + 2q−64 + 6q−66 + 2q−68−2q−70−3q−72 + 4q−74 + 7q−76 + 3q−78−3q−80−3q−82 + q−84 + 4q−86 + q−88−2q−90−3q−92−q−96−q−98−q−100 + q−104 + q−108

A3 Invariants.

Weight Invariant
0,1,0 1 + 3q−4 + 3q−6 + 3q−8 + 5q−10 + 3q−12−q−14−q−16−6q−18−8q−20−4q−22−3q−24−q−26 + 3q−28 + 5q−30 + 4q−32 + 2q−34 + q−36 + q−38−3q−40−q−42 + q−44−2q−46 + 2q−50−q−52−q−54 + q−56
1,0,0 q−1 + q−3 + 3q−5 + 2q−7 + 3q−9−q−13−3q−15−3q−17−2q−19−q−21 + q−23 + 2q−27 + q−31
1,0,1 q2 + 6q−2 + 2q−4 + 11q−6 + 8q−8 + 6q−10 + 12q−12−7q−14 + 7q−16−14q−18−8q−20−11q−22−19q−24−3q−26−13q−28 + 7q−30−3q−32 + 13q−34 + 7q−36 + 11q−38 + 9q−40 + q−42 + 10q−44−8q−46 + 7q−48−8q−50−2q−52−q−54−9q−56 + 4q−58−6q−60 + q−62 + 2q−64−2q−66 + 2q−68 + q−72 + 2q−74−q−76−q−78 + 2q−80−2q−82 + q−84 + q−86−2q−88 + q−90

A4 Invariants.

Weight Invariant
0,1,0,0 q−2 + q−4 + 3q−6 + 5q−8 + 6q−10 + 7q−12 + 8q−14 + 5q−16 + 2q−18−2q−20−7q−22−13q−24−14q−26−12q−28−9q−30−6q−32 + 4q−34 + 10q−36 + 9q−38 + 11q−40 + 10q−42 + 4q−44−q−46−q−48−4q−50−5q−52−3q−54−q−58−q−60 + 2q−62 + q−64−q−66 + q−70
1,0,0,0 q−2 + q−4 + 3q−6 + 3q−8 + 3q−10 + 3q−12−q−16−4q−18−3q−20−4q−22−2q−24−q−26 + q−28 + q−30 + q−32 + 2q−34 + q−38

B2 Invariants.

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

D4 Invariants.

Weight Invariant
1,0,0,0 q−2 + 3q−6 + q−8 + 6q−10 + 3q−12 + 6q−14 + 3q−16 + 5q−18−q−20−2q−22−5q−24−7q−26−6q−28−9q−30−3q−32−6q−34 + 3q−36−q−38 + 7q−40 + 2q−42 + 7q−44 + q−46 + 4q−48−q−54−2q−56−2q−60 + 2q−62−2q−64 + q−66−q−68 + 2q−70−q−72−q−76 + q−78

G2 Invariants.

Weight Invariant
1,0 q−2 + 3q−6−2q−8 + 2q−10 + q−12−2q−14 + 7q−16−5q−18 + 5q−20 + q−22−2q−24 + 8q−26−5q−28 + 5q−30 + 2q−32−2q−34 + 3q−36−3q−38 + 2q−42−4q−44 + 2q−46−4q−48−2q−50 + 2q−52−9q−54 + 4q−56−7q−58 + q−62−7q−64 + 7q−66−7q−68 + 4q−70 + 2q−72−5q−74 + 5q−76−2q−78 + q−80 + 4q−82−2q−84 + 3q−86 + q−88−q−90 + 4q−92−4q−94 + 3q−96−2q−100 + 3q−102−3q−104 + 3q−106−q−108 + q−110−3q−114 + 2q−116−2q−118 + 2q−120−q−122−q−128 + q−130−q−132 + q−134

.

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 5"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t3 + 3t2−4t + 5−4t−1 + 3t−2−t−3
In[5]:= Conway[K][z]
Out[5]= −z6−3z4−z2 + 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]= { 21, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q8−2q7 + 3q6−4q5 + 3q4−3q3 + 3q2−q + 1
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z6a−4 + z4a−2−5z4a−4 + z4a−6 + 4z2a−2−8z2a−4 + 3z2a−6 + 4a−2−5a−4 + 2a−6
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z7a−3 + z7a−5 + z6a−2 + 4z6a−4 + 3z6a−6−3z5a−3 + z5a−5 + 4z5a−7−5z4a−2−15z4a−4−7z4a−6 + 3z4a−8−10z3a−5−8z3a−7 + 2z3a−9 + 8z2a−2 + 15z2a−4 + 4z2a−6−2z2a−8 + z2a−10 + 3za−3 + 7za−5 + 4za−7−4a−2−5a−4−2a−6

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

Same Alexander/Conway Polynomial: {10_141,}

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 5"];
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 + 3t2−4t + 5−4t−1 + 3t−2−t−3, q8−2q7 + 3q6−4q5 + 3q4−3q3 + 3q2−q + 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]= {10_141,}
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: (-1, -3)

[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 8 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-10123456χ
17        11
15       1 -1
13      21 1
11     21  -1
9    12   -1
7   22    0
5  11     0
3 13      2
1         0
-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}_2 {\mathbb Z}
r = 0 {\mathbb Z}^{3} {\mathbb Z}
r = 1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 6 {\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 q22−2q21 + q20 + 2q19−6q18 + 5q17 + 3q16−10q15 + 7q14 + 5q13−12q12 + 6q11 + 7q10−12q9 + 3q8 + 8q7−9q6 + 7q4−5q3−q2 + 4q−1−q−1 + q−2
3 q42−2q41 + q40−q37 + 2q36−4q34 + 2q33 + 7q32−4q31−11q30 + 6q29 + 13q28−4q27−17q26 + 5q25 + 15q24−17q22−q21 + 13q20 + 6q19−12q18−8q17 + 10q16 + 9q15−6q14−13q13 + 6q12 + 11q11−14q9 + q8 + 9q7 + 6q6−11q5−4q4 + 5q3 + 7q2−4q−4 + 4q−2−q−4−q−5 + q−6
4 q68−2q67 + q66−2q64 + 5q63−4q62 + 2q61−3q60−3q59 + 14q58−7q57−2q56−10q55−3q54 + 32q53−3q52−12q51−29q50−7q49 + 56q48 + 9q47−15q46−49q45−20q44 + 68q43 + 21q42−7q41−55q40−33q39 + 65q38 + 21q37 + 4q36−44q35−38q34 + 54q33 + 12q32 + 12q31−30q30−37q29 + 40q28 + 3q27 + 20q26−17q25−36q24 + 25q23−4q22 + 27q21−4q20−32q19 + 11q18−14q17 + 28q16 + 11q15−21q14 + q13−23q12 + 18q11 + 18q10−4q9 + 3q8−26q7 + 2q6 + 12q5 + 6q4 + 10q3−17q2−6q + 1 + 4q−1 + 11q−2−5q−3−3q−4−3q−5−q−6 + 5q−7−q−10−q−11 + q−12
5 q100−2q99 + q98−2q96 + 3q95 + 2q94−4q93−q92 + q91 + 6q89 + q88−11q87−8q86 + 8q85 + 16q84 + 13q83−12q82−33q81−23q80 + 22q79 + 55q78 + 36q77−29q76−80q75−58q74 + 29q73 + 108q72 + 87q71−28q70−126q69−115q68 + 11q67 + 143q66 + 138q65 + 4q64−139q63−157q62−23q61 + 137q60 + 158q59 + 35q58−117q57−160q56−45q55 + 111q54 + 143q53 + 45q52−88q51−137q50−48q49 + 83q48 + 120q47 + 45q46−65q45−111q44−49q43 + 55q42 + 101q41 + 51q40−41q39−88q38−55q37 + 21q36 + 80q35 + 60q34−8q33−59q32−61q31−17q30 + 46q29 + 61q28 + 23q27−19q26−49q25−45q24 + 5q23 + 39q22 + 35q21 + 21q20−15q19−44q18−26q17 + 3q16 + 18q15 + 36q14 + 20q13−15q12−26q11−19q10−12q9 + 17q8 + 28q7 + 13q6−2q5−14q4−24q3−7q2 + 9q + 15 + 13q−1 + 4q−2−13q−3−11q−4−5q−5 + q−6 + 8q−7 + 9q−8−q−9−3q−10−3q−11−4q−12 + 4q−14 + q−15−q−18−q−19 + q−20
6 q138−2q137 + q136−2q134 + 3q133 + 2q131−7q130 + 3q129 + 4q128−5q127 + 5q126−2q125−3q124−11q123 + 15q122 + 16q121−9q120−3q119−19q118−19q117−5q116 + 55q115 + 52q114−17q113−40q112−76q111−55q110 + 20q109 + 142q108 + 135q107−12q106−107q105−190q104−144q103 + 34q102 + 264q101 + 279q100 + 50q99−155q98−327q97−294q96−22q95 + 342q94 + 432q93 + 175q92−118q91−401q90−437q89−136q88 + 324q87 + 503q86 + 280q85−27q84−376q83−483q82−223q81 + 251q80 + 477q79 + 298q78 + 36q77−307q76−437q75−237q74 + 192q73 + 413q72 + 256q71 + 52q70−250q69−368q68−221q67 + 155q66 + 356q65 + 216q64 + 65q63−204q62−315q61−222q60 + 109q59 + 303q58 + 200q57 + 105q56−144q55−268q54−242q53 + 35q52 + 229q51 + 187q50 + 161q49−57q48−199q47−254q46−51q45 + 127q44 + 144q43 + 197q42 + 44q41−93q40−224q39−116q38 + 12q37 + 58q36 + 178q35 + 116q34 + 29q33−139q32−118q31−71q30−47q29 + 95q28 + 114q27 + 112q26−29q25−49q24−75q23−108q22−9q21 + 39q20 + 108q19 + 37q18 + 37q17−10q16−85q15−58q14−40q13 + 39q12 + 19q11 + 63q10 + 49q9−13q8−29q7−52q6−14q5−32q4 + 24q3 + 43q2 + 26q + 15−12q−1−8q−2−44q−3−12q−4 + 5q−5 + 13q−6 + 18q−7 + 12q−8 + 15q−9−19q−10−11q−11−9q−12−4q−13 + q−14 + 6q−15 + 14q−16−2q−17−3q−19−3q−20−4q−21−q−22 + 5q−23 + q−25−q−28−q−29 + q−30
7 q182−2q181 + q180−2q178 + 3q177−q174−3q173 + 6q172−q171−6q170 + 5q169−3q168 + 2q166 + q165 + 13q164−4q163−19q162−7q161−9q160 + 14q159 + 29q158 + 18q157 + 17q156−31q155−66q154−47q153−11q152 + 77q151 + 119q150 + 77q149 + 4q148−131q147−203q146−146q145 + 226q143 + 332q142 + 233q141 + 2q140−328q139−488q138−374q137−40q136 + 427q135 + 679q134 + 562q133 + 119q132−503q131−868q130−766q129−256q128 + 507q127 + 1020q126 + 996q125 + 439q124−466q123−1113q122−1177q121−631q120 + 349q119 + 1131q118 + 1308q117 + 807q116−211q115−1082q114−1363q113−937q112 + 85q111 + 999q110 + 1339q109 + 988q108 + 24q107−888q106−1286q105−999q104−69q103 + 815q102 + 1191q101 + 942q100 + 91q99−736q98−1108q97−900q96−79q95 + 707q94 + 1031q93 + 828q92 + 68q91−655q90−969q89−803q88−71q87 + 627q86 + 920q85 + 770q84 + 94q83−564q82−865q81−772q80−144q79 + 495q78 + 808q77 + 772q76 + 207q75−399q74−734q73−770q72−288q71 + 280q70 + 648q69 + 767q68 + 367q67−163q66−533q65−728q64−445q63 + 16q62 + 406q61 + 688q60 + 494q59 + 106q58−252q57−590q56−527q55−239q54 + 100q53 + 483q52 + 507q51 + 320q50 + 65q49−326q48−454q47−384q46−200q45 + 179q44 + 346q43 + 371q42 + 304q41−3q40−214q39−332q38−351q37−109q36 + 66q35 + 209q34 + 336q33 + 212q32 + 71q31−101q30−271q29−213q28−158q27−43q26 + 153q25 + 191q24 + 202q23 + 126q22−49q21−99q20−168q19−176q18−54q17 + 10q16 + 112q15 + 162q14 + 91q13 + 66q12−23q11−108q10−94q9−104q8−39q7 + 43q6 + 52q5 + 98q4 + 73q3 + 13q2−5q−60−70q−1−41q−2−36q−3 + 22q−4 + 46q−5 + 35q−6 + 47q−7 + 15q−8−11q−9−21q−10−46q−11−23q−12−4q−13−2q−14 + 21q−15 + 22q−16 + 18q−17 + 14q−18−13q−19−12q−20−8q−21−13q−22−4q−23 + q−24 + 7q−25 + 13q−26 + q−27 + q−29−4q−30−3q−31−4q−32−q−33 + 4q−34 + q−35 + q−37−q−40−q−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.

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