10 69

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10_68

10_70

Contents

Image:10 69.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 69's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_69's page at Knotilus!

Visit 10 69's page at the original Knot Atlas!

[edit 10 69 Quick Notes]
[edit 10 69 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8
Gauss code -1, 4, -3, 1, -6, 7, -2, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8
Dowker-Thistlethwaite code 4 10 14 12 18 2 16 6 20 8
Conway Notation [211,21,21]


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

Length is 12, width is 5,

Braid index is 5

Image:10 69_ML.gif Image:10 69_AP.gif
[{15, 3}, {4, 2}, {3, 7}, {1, 4}, {6, 13}, {8, 10}, {7, 9}, {5, 8}, {2, 6}, {14, 11}, {10, 12}, {9, 5}, {11, 1}, {13, 15}, {12, 14}]

[edit Notes on presentations of 10 69]

Knot 10_69.
Knot 10_69.
A graph, knot 10_69.
A graph, knot 10_69.
A part of a knot and a part of a graph.
A part of a knot and a part of a graph.
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["10 69"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, -6, 7, -2, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8
In[6]:= DTCode[K]
Out[6]= 4 10 14 12 18 2 16 6 20 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [211,21,21]
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(5,{1,1,2,−1,−3,2,1,4,−3,2,−3,4})
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]= { 5, 12, 5 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.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:10 69_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{15, 3}, {4, 2}, {3, 7}, {1, 4}, {6, 13}, {8, 10}, {7, 9}, {5, 8}, {2, 6}, {14, 11}, {10, 12}, {9, 5}, {11, 1}, {13, 15}, {12, 14}]
In[14]:= Draw[ap]
Image:10 69_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 2
Maximal Thurston-Bennequin number [-2][-10]
Hyperbolic Volume 14.1265
A-Polynomial See Data:10 69/A-polynomial

[edit Notes for 10 69'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 10 69's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t3−7t2 + 21t−29 + 21t−1−7t−2 + t−3
Conway polynomial z6−z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 87, 2 }
Jones polynomial −q8 + 3q7−7q6 + 11q5−13q4 + 15q3−14q2 + 11q−7 + 4q−1−q−2
HOMFLY-PT polynomial (db, data sources) z6a−2 + 3z4a−2−3z4a−4−z4 + 5z2a−2−5z2a−4 + 3z2a−6−z2 + 2a−2−2a−4 + 2a−6−a−8
Kauffman polynomial (db, data sources) z9a−3 + z9a−5 + 4z8a−2 + 8z8a−4 + 4z8a−6 + 6z7a−1 + 14z7a−3 + 13z7a−5 + 5z7a−7 + 3z6a−2−4z6a−4 + 3z6a−8 + 4z6 + az5−10z5a−1−32z5a−3−30z5a−5−8z5a−7 + z5a−9−17z4a−2−14z4a−4−9z4a−6−5z4a−8−7z4−az3 + 5z3a−1 + 22z3a−3 + 23z3a−5 + 5z3a−7−2z3a−9 + 11z2a−2 + 12z2a−4 + 7z2a−6 + 3z2a−8 + 3z2−za−1−4za−3−6za−5−2za−7 + za−9−2a−2−2a−4−2a−6−a−8
The A2 invariant −q6 + 2q4−q2 + 4q−2−3q−4 + 2q−6−q−8 + 2q−12−2q−14 + 3q−16−q−18−q−20 + 2q−22−q−24−q−26
The G2 invariant q32−3q30 + 7q28−13q26 + 14q24−12q22−q20 + 26q18−51q16 + 77q14−84q12 + 57q10−q8−83q6 + 165q4−206q2 + 191−107q−2−23q−4 + 170q−6−269q−8 + 282q−10−199q−12 + 45q−14 + 111q−16−215q−18 + 217q−20−120q−22−17q−24 + 156q−26−210q−28 + 145q−30 + 6q−32−193q−34 + 324q−36−341q−38 + 228q−40−18q−42−207q−44 + 382q−46−429q−48 + 337q−50−147q−52−86q−54 + 260q−56−320q−58 + 260q−60−108q−62−54q−64 + 173q−66−190q−68 + 100q−70 + 42q−72−183q−74 + 249q−76−204q−78 + 69q−80 + 100q−82−232q−84 + 292q−86−249q−88 + 132q−90 + 7q−92−133q−94 + 191q−96−182q−98 + 127q−100−49q−102−15q−104 + 55q−106−69q−108 + 56q−110−35q−112 + 14q−114 + q−116−9q−118 + 9q−120−8q−122 + 5q−124−2q−126 + q−128
Further Quantum Invariants
Further quantum knot invariants for 10_69.

A1 Invariants.

Weight Invariant
1 −q5 + 3q3−3q + 4q−1−3q−3 + q−5 + 2q−7−2q−9 + 4q−11−4q−13 + 2q−15−q−17
2 q16−3q14−q12 + 11q10−9q8−11q6 + 27q4−10q2−29 + 37q−2 + 3q−4−37q−6 + 26q−8 + 17q−10−25q−12−3q−14 + 18q−16 + 2q−18−28q−20 + 12q−22 + 29q−24−36q−26−2q−28 + 38q−30−26q−32−13q−34 + 26q−36−8q−38−10q−40 + 8q−42−2q−46 + q−48
3 −q33 + 3q31 + q29−7q27−6q25 + 13q23 + 21q21−24q19−40q17 + 27q15 + 76q13−24q11−122q9 + 2q7 + 171q5 + 40q3−208q−101q−1 + 227q−3 + 170q−5−212q−7−227q−9 + 166q−11 + 266q−13−105q−15−268q−17 + 31q−19 + 242q−21 + 48q−23−195q−25−109q−27 + 128q−29 + 167q−31−57q−33−212q−35−22q−37 + 238q−39 + 98q−41−245q−43−165q−45 + 220q−47 + 223q−49−175q−51−250q−53 + 110q−55 + 251q−57−48q−59−215q−61−11q−63 + 166q−65 + 42q−67−110q−69−47q−71 + 57q−73 + 41q−75−25q−77−26q−79 + 9q−81 + 12q−83−2q−85−4q−87 + 2q−91−q−93
4 q56−3q54−q52 + 7q50 + 2q48 + 2q46−23q44−13q42 + 33q40 + 29q38 + 28q36−89q34−94q32 + 65q30 + 139q28 + 164q26−193q24−350q22−35q20 + 333q18 + 593q16−131q14−795q12−538q10 + 349q8 + 1324q6 + 445q4−1054q2−1455−251q−2 + 1853q−4 + 1509q−6−591q−8−2159q−10−1370q−12 + 1556q−14 + 2328q−16 + 480q−18−1991q−20−2216q−22 + 544q−24 + 2241q−26 + 1394q−28−1067q−30−2215q−32−522q−34 + 1412q−36 + 1715q−38 + 15q−40−1576q−42−1258q−44 + 372q−46 + 1604q−48 + 963q−50−712q−52−1757q−54−672q−56 + 1285q−58 + 1781q−60 + 265q−62−1979q−64−1672q−66 + 632q−68 + 2260q−70 + 1348q−72−1610q−74−2315q−76−389q−78 + 1986q−80 + 2133q−82−609q−84−2128q−86−1280q−88 + 975q−90 + 2070q−92 + 394q−94−1173q−96−1402q−98−53q−100 + 1241q−102 + 719q−104−216q−106−843q−108−422q−110 + 403q−112 + 434q−114 + 157q−116−265q−118−269q−120 + 37q−122 + 118q−124 + 117q−126−33q−128−80q−130−9q−132 + 8q−134 + 31q−136 + q−138−13q−140−2q−144 + 4q−146−2q−150 + q−152
5 −q85 + 3q83 + q81−7q79−2q77 + 2q75 + 8q73 + 15q71 + 4q69−33q67−43q65−q63 + 58q61 + 95q59 + 42q57−98q55−226q53−139q51 + 167q49 + 419q47 + 351q45−136q43−745q41−823q39−7q37 + 1144q35 + 1555q33 + 546q31−1453q29−2717q27−1638q25 + 1465q23 + 4103q21 + 3502q19−723q17−5462q15−6166q13−1115q11 + 6247q9 + 9339q7 + 4252q5−5844q3−12391q−8571q−1 + 3809q−3 + 14563q−5 + 13421q−7−96q−9−15055q−11−17894q−13−4872q−15 + 13525q−17 + 21057q−19 + 10203q−21−10166q−23−22157q−25−14879q−27 + 5525q−29 + 21093q−31 + 18124q−33−590q−35−18204q−37−19422q−39−3902q−41 + 14133q−43 + 18999q−45 + 7395q−47−9700q−49−17272q−51−9699q−53 + 5399q−55 + 14814q−57 + 11177q−59−1575q−61−12261q−63−12095q−65−1803q−67 + 9744q−69 + 12940q−71 + 5020q−73−7435q−75−13829q−77−8277q−79 + 5020q−81 + 14706q−83 + 11762q−85−2198q−87−15282q−89−15339q−91−1250q−93 + 15027q−95 + 18637q−97 + 5399q−99−13533q−101−21098q−103−9887q−105 + 10536q−107 + 22050q−109 + 14178q−111−6236q−113−21050q−115−17417q−117 + 1110q−119 + 18082q−121 + 18967q−123 + 3828q−125−13500q−127−18371q−129−7804q−131 + 8192q−133 + 15916q−135 + 9953q−137−3176q−139−12057q−141−10227q−143−723q−145 + 7853q−147 + 8911q−149 + 2955q−151−4092q−153−6647q−155−3682q−157 + 1358q−159 + 4271q−161 + 3311q−163 + 145q−165−2274q−167−2393q−169−747q−171 + 955q−173 + 1466q−175 + 753q−177−270q−179−745q−181−529q−183−11q−185 + 314q−187 + 304q−189 + 75q−191−119q−193−139q−195−50q−197 + 30q−199 + 49q−201 + 34q−203−7q−205−24q−207−6q−209 + 3q−211 + q−213 + 4q−215 + 2q−217−4q−219 + 2q−223−q−225

A2 Invariants.

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

A3 Invariants.

Weight Invariant
0,1,0 q14−3q12 + q10 + 6q8−12q6 + 5q4 + 16q2−23 + 6q−2 + 25q−4−30q−6 + 2q−8 + 27q−10−21q−12−4q−14 + 16q−16−q−18−8q−20−3q−22 + 19q−24−6q−26−21q−28 + 27q−30−30q−34 + 25q−36 + 3q−38−23q−40 + 15q−42 + 2q−44−10q−46 + 5q−48 + q−50−2q−52 + q−54
1,0,0 −q7 + 2q5−2q3 + 2q−q−1 + 4q−3−2q−5 + 2q−7−q−15 + 2q−17−3q−19 + 3q−21−q−23 + 2q−25−q−27 + 2q−29−q−31−q−33−q−35

B2 Invariants.

Weight Invariant
0,1 −q14 + 3q12−7q10 + 12q8−18q6 + 25q4−30q2 + 35−34q−2 + 31q−4−20q−6 + 8q−8 + 9q−10−27q−12 + 44q−14−58q−16 + 65q−18−68q−20 + 63q−22−53q−24 + 38q−26−19q−28 + 3q−30 + 14q−32−24q−34 + 33q−36−35q−38 + 33q−40−29q−42 + 22q−44−16q−46 + 9q−48−5q−50 + 2q−52−q−54
1,0 q24−3q20−3q18 + 4q16 + 9q14−q12−15q10−9q8 + 17q6 + 23q4−6q2−32−11q−2 + 30q−4 + 29q−6−17q−8−37q−10−q−12 + 35q−14 + 15q−16−25q−18−20q−20 + 17q−22 + 23q−24−10q−26−23q−28 + 5q−30 + 24q−32−24q−36−5q−38 + 25q−40 + 14q−42−24q−44−23q−46 + 18q−48 + 34q−50−5q−52−38q−54−15q−56 + 30q−58 + 28q−60−14q−62−31q−64−4q−66 + 22q−68 + 13q−70−9q−72−13q−74 + 7q−78 + 3q−80−2q−82−2q−84 + q−88

G2 Invariants.

Weight Invariant
1,0 q32−3q30 + 7q28−13q26 + 14q24−12q22−q20 + 26q18−51q16 + 77q14−84q12 + 57q10−q8−83q6 + 165q4−206q2 + 191−107q−2−23q−4 + 170q−6−269q−8 + 282q−10−199q−12 + 45q−14 + 111q−16−215q−18 + 217q−20−120q−22−17q−24 + 156q−26−210q−28 + 145q−30 + 6q−32−193q−34 + 324q−36−341q−38 + 228q−40−18q−42−207q−44 + 382q−46−429q−48 + 337q−50−147q−52−86q−54 + 260q−56−320q−58 + 260q−60−108q−62−54q−64 + 173q−66−190q−68 + 100q−70 + 42q−72−183q−74 + 249q−76−204q−78 + 69q−80 + 100q−82−232q−84 + 292q−86−249q−88 + 132q−90 + 7q−92−133q−94 + 191q−96−182q−98 + 127q−100−49q−102−15q−104 + 55q−106−69q−108 + 56q−110−35q−112 + 14q−114 + q−116−9q−118 + 9q−120−8q−122 + 5q−124−2q−126 + q−128

.

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["10 69"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t3−7t2 + 21t−29 + 21t−1−7t−2 + t−3
In[5]:= Conway[K][z]
Out[5]= z6−z4 + 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]= { 87, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + 3q7−7q6 + 11q5−13q4 + 15q3−14q2 + 11q−7 + 4q−1−q−2
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z6a−2 + 3z4a−2−3z4a−4−z4 + 5z2a−2−5z2a−4 + 3z2a−6−z2 + 2a−2−2a−4 + 2a−6−a−8
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z9a−3 + z9a−5 + 4z8a−2 + 8z8a−4 + 4z8a−6 + 6z7a−1 + 14z7a−3 + 13z7a−5 + 5z7a−7 + 3z6a−2−4z6a−4 + 3z6a−8 + 4z6 + az5−10z5a−1−32z5a−3−30z5a−5−8z5a−7 + z5a−9−17z4a−2−14z4a−4−9z4a−6−5z4a−8−7z4−az3 + 5z3a−1 + 22z3a−3 + 23z3a−5 + 5z3a−7−2z3a−9 + 11z2a−2 + 12z2a−4 + 7z2a−6 + 3z2a−8 + 3z2−za−1−4za−3−6za−5−2za−7 + za−9−2a−2−2a−4−2a−6−a−8

[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["10 69"];
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]= { t3−7t2 + 21t−29 + 21t−1−7t−2 + t−3, −q8 + 3q7−7q6 + 11q5−13q4 + 15q3−14q2 + 11q−7 + 4q−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]= {}
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, 4)
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
8 32 32 \frac{412}{3} \frac{68}{3} 256 \frac{1952}{3} \frac{320}{3} 128 \frac{256}{3} 512 \frac{3296}{3} \frac{544}{3} \frac{45151}{15} -\frac{4844}{15} \frac{75004}{45} \frac{689}{9} \frac{3631}{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 = 2 is the signature of 10 69. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-101234567χ
17          1-1
15         2 2
13        51 -4
11       62  4
9      75   -2
7     86    2
5    67     1
3   58      -3
1  37       4
-1 14        -3
-3 3         3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r = 1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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 q23−3q22 + 2q21 + 9q20−21q19 + 4q18 + 43q17−60q16−9q15 + 107q14−100q13−43q12 + 172q11−117q10−83q9 + 202q8−101q7−104q6 + 180q5−59q4−95q3 + 117q2−19q−61 + 51q−1−24q−3 + 13q−4 + 2q−5−4q−6 + q−7
3 −q45 + 3q44−2q43−4q42 + q41 + 17q40−5q39−39q38 + 2q37 + 83q36 + 11q35−143q34−61q33 + 235q32 + 135q31−320q30−265q29 + 402q28 + 434q27−461q26−625q25 + 477q24 + 832q23−464q22−1010q21 + 397q20 + 1175q19−324q18−1270q17 + 207q16 + 1330q15−100q14−1309q13−30q12 + 1244q11 + 143q10−1115q9−241q8 + 945q7 + 306q6−744q5−341q4 + 552q3 + 321q2−362q−284 + 224q−1 + 214q−2−114q−3−153q−4 + 55q−5 + 90q−6−16q−7−53q−8 + 6q−9 + 23q−10−8q−12−2q−13 + 4q−14−q−15
4 q74−3q73 + 2q72 + 4q71−6q70 + 3q69−16q68 + 16q67 + 34q66−29q65−14q64−87q63 + 63q62 + 184q61−28q60−95q59−393q58 + 67q57 + 606q56 + 249q55−126q54−1218q53−354q52 + 1233q51 + 1184q50 + 396q49−2512q48−1703q47 + 1462q46 + 2751q45 + 2072q44−3607q43−3958q42 + 614q41 + 4270q40 + 4814q39−3754q38−6333q37−1312q36 + 4975q35 + 7772q34−2842q33−7961q32−3616q31 + 4668q30 + 10016q29−1326q28−8457q27−5573q26 + 3583q25 + 11061q24 + 349q23−7816q22−6805q21 + 1953q20 + 10743q19 + 1940q18−6116q17−7108q16 + 19q15 + 9050q14 + 3088q13−3655q12−6261q11−1678q10 + 6290q9 + 3313q8−1184q7−4413q6−2450q5 + 3364q4 + 2524q3 + 384q2−2313q−2106 + 1260q−1 + 1320q−2 + 785q−3−814q−4−1227q−5 + 285q−6 + 433q−7 + 528q−8−150q−9−503q−10 + 25q−11 + 65q−12 + 213q−13 + 7q−14−146q−15−9q−17 + 54q−18 + 12q−19−29q−20 + q−21−5q−22 + 8q−23 + 2q−24−4q−25 + q−26
5 −q110 + 3q109−2q108−4q107 + 6q106 + 2q105−4q104 + 5q103−11q102−22q101 + 23q100 + 43q99 + 11q98−14q97−91q96−111q95 + 43q94 + 237q93 + 240q92−4q91−416q90−629q89−173q88 + 712q87 + 1263q86 + 709q85−927q84−2331q83−1819q82 + 831q81 + 3682q80 + 3875q79 + 33q78−5244q77−6859q76−2134q75 + 6237q74 + 10922q73 + 5989q72−6302q71−15435q70−11638q69 + 4407q68 + 19803q67 + 19118q66−339q65−23159q64−27634q63−6160q62 + 24674q61 + 36446q60 + 14800q59−24044q58−44606q57−24687q56 + 21041q55 + 51260q54 + 35214q53−16172q52−56120q51−45110q50 + 9830q49 + 58825q48 + 54146q47−2934q46−59730q45−61387q44−4259q43 + 58882q42 + 67230q41 + 11026q40−56786q39−71073q38−17556q37 + 53330q36 + 73624q35 + 23481q34−48866q33−74269q32−29103q31 + 43038q30 + 73458q29 + 34167q28−36114q27−70632q26−38518q25 + 27940q24 + 65885q23 + 41739q22−19019q21−59028q20−43384q19 + 9905q18 + 50365q17 + 42957q16−1405q15−40314q14−40415q13−5663q12 + 29961q11 + 35679q10 + 10586q9−19945q8−29561q7−13195q6 + 11564q5 + 22657q4 + 13425q3−4986q2−16044q−12053 + 810q−1 + 10277q−2 + 9605q−3 + 1561q−4−5948q−5−6966q−6−2282q−7 + 2915q−8 + 4554q−9 + 2265q−10−1209q−11−2741q−12−1681q−13 + 277q−14 + 1451q−15 + 1186q−16 + 55q−17−742q−18−672q−19−134q−20 + 300q−21 + 370q−22 + 133q−23−133q−24−185q−25−66q−26 + 48q−27 + 64q−28 + 46q−29−5q−30−45q−31−13q−32 + 11q−33 + 5q−34 + 4q−35 + 5q−36−8q−37−2q−38 + 4q−39−q−40
6 q153−3q152 + 2q151 + 4q150−6q149−2q148−q147 + 15q146−10q145−q144 + 28q143−37q142−30q141−13q140 + 77q139 + 25q138 + 16q137 + 95q136−172q135−228q134−159q133 + 263q132 + 308q131 + 360q130 + 480q129−561q128−1153q127−1254q126 + 185q125 + 1187q124 + 2222q123 + 2830q122−345q121−3582q120−5840q119−3119q118 + 999q117 + 7049q116 + 11702q115 + 5902q114−4632q113−16430q112−17035q111−9338q110 + 9969q109 + 30667q108 + 29753q107 + 9643q106−25339q105−46128q104−46371q103−9077q102 + 48021q101 + 76748q100 + 61052q99−5489q98−73901q97−115855q96−76895q95 + 29385q94 + 125127q93 + 154418q92 + 73981q91−59584q90−190230q89−196007q88−58377q87 + 128868q86 + 256318q85 + 212139q84 + 29782q83−219573q82−327261q81−209585q80 + 56438q79 + 314426q78 + 365081q77 + 183183q76−174563q75−417104q74−376572q73−77438q72 + 301775q71 + 479632q70 + 351254q69−71583q68−440451q67−507943q66−225474q65 + 234221q64 + 532645q63 + 486687q62 + 47351q61−410722q60−582966q59−347730q58 + 146413q57 + 534670q56 + 572713q55 + 151582q54−354261q53−609988q52−433462q51 + 59356q50 + 504456q49 + 616966q48 + 236940q47−282948q46−601618q45−491012q44−28635q43 + 446200q42 + 627244q41 + 312203q40−190207q39−555571q38−523805q37−125675q36 + 349536q35 + 595214q34 + 375256q33−69685q32−458164q31−516598q30−222600q29 + 210606q28 + 503619q27 + 402191q26 + 61232q25−308130q24−447067q23−285746q22 + 54973q21 + 353221q20 + 364485q19 + 159174q18−138309q17−316308q16−280859q15−66294q14 + 182376q13 + 263001q12 + 185928q11−4633q10−165324q9−209517q8−114346q7 + 48543q6 + 139926q5 + 145060q4 + 56360q3−49406q2−114531q−96180−16455q−1 + 46574q−2 + 79144q−3 + 55332q−4 + 5971q−5−42476q−6−53117q−7−26252q−8 + 2555q−9 + 29120q−10 + 30777q−11 + 16168q−12−8288q−13−19992q−14−15023q−15−7012q−16 + 6080q−17 + 11202q−18 + 9900q−19 + 788q−20−4943q−21−5119q−22−4545q−23−53q−24 + 2616q−25 + 3823q−26 + 1091q−27−702q−28−1015q−29−1647q−30−516q−31 + 305q−32 + 1102q−33 + 354q−34−35q−35−55q−36−407q−37−197q−38−30q−39 + 268q−40 + 56q−41−3q−42 + 32q−43−74q−44−40q−45−25q−46 + 59q−47 + 4q−48−10q−49 + 13q−50−10q−51−4q−52−5q−53 + 8q−54 + 2q−55−4q−56 + q−57

[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.

10_68

10_70

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