10 81

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10_80

10_82

Contents

Image:10 81.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 81's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_81's page at Knotilus!

Visit 10 81's page at the original Knot Atlas!

[edit 10 81 Quick Notes]
[edit 10 81 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X8493 X12,6,13,5 X16,9,17,10 X20,17,1,18 X18,13,19,14 X14,19,15,20 X10,15,11,16 X6,12,7,11 X2837
Gauss code 1, -10, 2, -1, 3, -9, 10, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -5
Dowker-Thistlethwaite code 4 8 12 2 16 6 18 10 20 14
Conway Notation [(21,2)(21,2)]


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

Length is 12, width is 5,

Braid index is 5

Image:10 81_ML.gif Image:10 81_AP.gif
[{3, 12}, {2, 5}, {1, 3}, {13, 9}, {12, 2}, {4, 7}, {6, 8}, {7, 10}, {9, 11}, {10, 6}, {5, 13}, {11, 4}, {8, 1}]

[edit Notes on presentations of 10 81]


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 81"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X8493 X12,6,13,5 X16,9,17,10 X20,17,1,18 X18,13,19,14 X14,19,15,20 X10,15,11,16 X6,12,7,11 X2837
In[5]:= GaussCode[K]
Out[5]= 1, -10, 2, -1, 3, -9, 10, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -5
In[6]:= DTCode[K]
Out[6]= 4 8 12 2 16 6 18 10 20 14

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [(21,2)(21,2)]
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(5,{1,1,−2,1,3,2,2,−4,−3,−3,−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:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.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:10 81_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 12}, {2, 5}, {1, 3}, {13, 9}, {12, 2}, {4, 7}, {6, 8}, {7, 10}, {9, 11}, {10, 6}, {5, 13}, {11, 4}, {8, 1}]
In[14]:= Draw[ap]
Image:10 81_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Negative amphicheiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 14.4927
A-Polynomial See Data:10 81/A-polynomial

[edit Notes for 10 81's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for 10 81's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t3 + 8t2−20t + 27−20t−1 + 8t−2−t−3
Conway polynomial −z6 + 2z4 + 3z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 85, 0 }
Jones polynomial −q5 + 3q4−7q3 + 11q2−13q + 15−13q−1 + 11q−2−7q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) −z6 + 2a2z4 + 2z4a−2−2z4−a4z2 + 3a2z2 + 3z2a−2−z2a−4−z2−a4 + a2 + a−2−a−4 + 1
Kauffman polynomial (db, data sources) az9 + z9a−1 + 4a2z8 + 4z8a−2 + 8z8 + 5a3z7 + 13az7 + 13z7a−1 + 5z7a−3 + 3a4z6 + 3z6a−4−6z6 + a5z5−8a3z5−31az5−31z5a−1−8z5a−3 + z5a−5−5a4z4−9a2z4−9z4a−2−5z4a−4−8z4−2a5z3 + 5a3z3 + 25az3 + 25z3a−1 + 5z3a−3−2z3a−5 + 3a4z2 + 6a2z2 + 6z2a−2 + 3z2a−4 + 6z2 + a5z−2a3z−8az−8za−1−2za−3 + za−5−a4−a2−a−2−a−4 + 1
The A2 invariant −q16 + q12−3q10 + 2q8−q4 + 4q2−1 + 4q−2−q−4 + 2q−8−3q−10 + q−12−q−16
The G2 invariant q80−2q78 + 5q76−8q74 + 9q72−9q70 + q68 + 14q66−35q64 + 56q62−69q60 + 55q58−16q56−50q54 + 129q52−183q50 + 191q48−130q46 + 4q44 + 139q42−255q40 + 293q38−227q36 + 79q34 + 91q32−219q30 + 247q28−166q26 + 14q24 + 137q22−214q20 + 173q18−34q16−147q14 + 296q12−335q10 + 249q8−55q6−172q4 + 360q2−427 + 360q−2−172q−4−55q−6 + 249q−8−335q−10 + 296q−12−147q−14−34q−16 + 173q−18−214q−20 + 137q−22 + 14q−24−166q−26 + 247q−28−219q−30 + 91q−32 + 79q−34−227q−36 + 293q−38−255q−40 + 139q−42 + 4q−44−130q−46 + 191q−48−183q−50 + 129q−52−50q−54−16q−56 + 55q−58−69q−60 + 56q−62−35q−64 + 14q−66 + q−68−9q−70 + 9q−72−8q−74 + 5q−76−2q−78 + q−80
Further Quantum Invariants
Further quantum knot invariants for 10_81.

A1 Invariants.

Weight Invariant
1 −q11 + 2q9−4q7 + 4q5−2q3 + 2q + 2q−1−2q−3 + 4q−5−4q−7 + 2q−9−q−11
2 q32−2q30 + 8q26−10q24−8q22 + 26q20−13q18−27q16 + 38q14−38q10 + 26q8 + 15q6−26q4 + q2 + 21 + q−2−26q−4 + 15q−6 + 26q−8−38q−10 + 38q−14−27q−16−13q−18 + 26q−20−8q−22−10q−24 + 8q−26−2q−30 + q−32
3 −q63 + 2q61−4q57−2q55 + 12q53 + 9q51−26q49−25q47 + 41q45 + 58q43−47q41−113q39 + 41q37 + 173q35−3q33−226q31−68q29 + 260q27 + 140q25−250q23−215q21 + 207q19 + 260q17−137q15−277q13 + 64q11 + 255q9 + 21q7−215q5−91q3 + 159q + 159q−1−91q−3−215q−5 + 21q−7 + 255q−9 + 64q−11−277q−13−137q−15 + 260q−17 + 207q−19−215q−21−250q−23 + 140q−25 + 260q−27−68q−29−226q−31−3q−33 + 173q−35 + 41q−37−113q−39−47q−41 + 58q−43 + 41q−45−25q−47−26q−49 + 9q−51 + 12q−53−2q−55−4q−57 + 2q−61−q−63
4 q104−2q102 + 4q98−2q96−13q92 + q90 + 31q88 + 8q86−9q84−80q82−34q80 + 117q78 + 121q76 + 39q74−273q72−277q70 + 153q68 + 456q66 + 436q64−420q62−907q60−292q58 + 744q56 + 1385q54 + 71q52−1490q50−1436q48 + 256q46 + 2270q44 + 1356q42−1174q40−2491q38−1058q36 + 2127q34 + 2496q32 + 51q30−2484q28−2193q26 + 1010q24 + 2579q22 + 1212q20−1509q18−2375q16−229q14 + 1776q12 + 1719q10−323q8−1855q6−1131q4 + 741q2 + 1799 + 741q−2−1131q−4−1855q−6−323q−8 + 1719q−10 + 1776q−12−229q−14−2375q−16−1509q−18 + 1212q−20 + 2579q−22 + 1010q−24−2193q−26−2484q−28 + 51q−30 + 2496q−32 + 2127q−34−1058q−36−2491q−38−1174q−40 + 1356q−42 + 2270q−44 + 256q−46−1436q−48−1490q−50 + 71q−52 + 1385q−54 + 744q−56−292q−58−907q−60−420q−62 + 436q−64 + 456q−66 + 153q−68−277q−70−273q−72 + 39q−74 + 121q−76 + 117q−78−34q−80−80q−82−9q−84 + 8q−86 + 31q−88 + q−90−13q−92−2q−96 + 4q−98−2q−102 + q−104
5 −q155 + 2q153−4q149 + 2q147 + 4q145 + q143 + 3q141−6q139−24q137−7q135 + 34q133 + 49q131 + 30q129−49q127−139q125−122q123 + 73q121 + 307q119 + 323q117−3q115−536q113−776q111−304q109 + 763q107 + 1544q105 + 1063q103−721q101−2541q99−2552q97−33q95 + 3502q93 + 4831q91 + 1897q89−3746q87−7535q85−5268q83 + 2547q81 + 9987q79 + 9878q77 + 624q75−11060q73−14906q71−5906q69 + 9898q67 + 19206q65 + 12485q63−6200q61−21361q59−19110q57 + 243q55 + 20810q53 + 24323q51 + 6647q49−17440q47−26957q45−13219q43 + 12096q41 + 26728q39 + 18078q37−5902q35−23973q33−20683q31 + 113q29 + 19538q27 + 21025q25 + 4588q23−14499q21−19760q19−7842q17 + 9640q15 + 17570q13 + 10084q11−5387q9−15327q7−11708q5 + 1734q3 + 13340q + 13340q−1 + 1734q−3−11708q−5−15327q−7−5387q−9 + 10084q−11 + 17570q−13 + 9640q−15−7842q−17−19760q−19−14499q−21 + 4588q−23 + 21025q−25 + 19538q−27 + 113q−29−20683q−31−23973q−33−5902q−35 + 18078q−37 + 26728q−39 + 12096q−41−13219q−43−26957q−45−17440q−47 + 6647q−49 + 24323q−51 + 20810q−53 + 243q−55−19110q−57−21361q−59−6200q−61 + 12485q−63 + 19206q−65 + 9898q−67−5906q−69−14906q−71−11060q−73 + 624q−75 + 9878q−77 + 9987q−79 + 2547q−81−5268q−83−7535q−85−3746q−87 + 1897q−89 + 4831q−91 + 3502q−93−33q−95−2552q−97−2541q−99−721q−101 + 1063q−103 + 1544q−105 + 763q−107−304q−109−776q−111−536q−113−3q−115 + 323q−117 + 307q−119 + 73q−121−122q−123−139q−125−49q−127 + 30q−129 + 49q−131 + 34q−133−7q−135−24q−137−6q−139 + 3q−141 + q−143 + 4q−145 + 2q−147−4q−149 + 2q−153−q−155

A2 Invariants.

Weight Invariant
1,0 −q16 + q12−3q10 + 2q8−q4 + 4q2−1 + 4q−2−q−4 + 2q−8−3q−10 + q−12−q−16
2,0 q42−2q38 + 5q34 + 3q32−8q30−5q28 + 11q26 + q24−17q22−6q20 + 17q18 + 5q16−20q14 + 3q12 + 18q10−5q8−10q6 + 9q4 + 6q2−6 + 6q−2 + 9q−4−10q−6−5q−8 + 18q−10 + 3q−12−20q−14 + 5q−16 + 17q−18−6q−20−17q−22 + q−24 + 11q−26−5q−28−8q−30 + 3q−32 + 5q−34−2q−38 + q−42

A3 Invariants.

Weight Invariant
0,1,0 q34−2q32 + q30 + 5q28−10q26 + 2q24 + 15q22−23q20 + 3q18 + 24q16−31q14 + 24q10−22q8−4q6 + 19q4 + 2q2−2 + 2q−2 + 19q−4−4q−6−22q−8 + 24q−10−31q−14 + 24q−16 + 3q−18−23q−20 + 15q−22 + 2q−24−10q−26 + 5q−28 + q−30−2q−32 + q−34
1,0,0 −q21−q17 + q15−3q13 + 3q11−2q9 + 2q7−q5 + 3q3 + q + q−1 + 3q−3−q−5 + 2q−7−2q−9 + 3q−11−3q−13 + q−15−q−17−q−21

B2 Invariants.

Weight Invariant
0,1 −q34 + 2q32−5q30 + 9q28−16q26 + 22q24−29q22 + 33q20−35q18 + 32q16−23q14 + 12q12 + 6q10−22q8 + 40q6−53q4 + 64q2−68 + 64q−2−53q−4 + 40q−6−22q−8 + 6q−10 + 12q−12−23q−14 + 32q−16−35q−18 + 33q−20−29q−22 + 22q−24−16q−26 + 9q−28−5q−30 + 2q−32−q−34
1,0 q56−2q52−2q50 + 3q48 + 7q46−13q42−9q40 + 13q38 + 22q36−4q34−31q32−14q30 + 28q28 + 29q26−16q24−38q22−4q20 + 34q18 + 15q16−25q14−22q12 + 16q10 + 24q8−6q6−22q4 + 5q2 + 27 + 5q−2−22q−4−6q−6 + 24q−8 + 16q−10−22q−12−25q−14 + 15q−16 + 34q−18−4q−20−38q−22−16q−24 + 29q−26 + 28q−28−14q−30−31q−32−4q−34 + 22q−36 + 13q−38−9q−40−13q−42 + 7q−46 + 3q−48−2q−50−2q−52 + q−56

G2 Invariants.

Weight Invariant
1,0 q80−2q78 + 5q76−8q74 + 9q72−9q70 + q68 + 14q66−35q64 + 56q62−69q60 + 55q58−16q56−50q54 + 129q52−183q50 + 191q48−130q46 + 4q44 + 139q42−255q40 + 293q38−227q36 + 79q34 + 91q32−219q30 + 247q28−166q26 + 14q24 + 137q22−214q20 + 173q18−34q16−147q14 + 296q12−335q10 + 249q8−55q6−172q4 + 360q2−427 + 360q−2−172q−4−55q−6 + 249q−8−335q−10 + 296q−12−147q−14−34q−16 + 173q−18−214q−20 + 137q−22 + 14q−24−166q−26 + 247q−28−219q−30 + 91q−32 + 79q−34−227q−36 + 293q−38−255q−40 + 139q−42 + 4q−44−130q−46 + 191q−48−183q−50 + 129q−52−50q−54−16q−56 + 55q−58−69q−60 + 56q−62−35q−64 + 14q−66 + q−68−9q−70 + 9q−72−8q−74 + 5q−76−2q−78 + q−80

.

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 81"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t3 + 8t2−20t + 27−20t−1 + 8t−2−t−3
In[5]:= Conway[K][z]
Out[5]= −z6 + 2z4 + 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]= { 85, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q5 + 3q4−7q3 + 11q2−13q + 15−13q−1 + 11q−2−7q−3 + 3q−4−q−5
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z6 + 2a2z4 + 2z4a−2−2z4−a4z2 + 3a2z2 + 3z2a−2−z2a−4−z2−a4 + a2 + a−2−a−4 + 1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= az9 + z9a−1 + 4a2z8 + 4z8a−2 + 8z8 + 5a3z7 + 13az7 + 13z7a−1 + 5z7a−3 + 3a4z6 + 3z6a−4−6z6 + a5z5−8a3z5−31az5−31z5a−1−8z5a−3 + z5a−5−5a4z4−9a2z4−9z4a−2−5z4a−4−8z4−2a5z3 + 5a3z3 + 25az3 + 25z3a−1 + 5z3a−3−2z3a−5 + 3a4z2 + 6a2z2 + 6z2a−2 + 3z2a−4 + 6z2 + a5z−2a3z−8az−8za−1−2za−3 + za−5−a4−a2−a−2−a−4 + 1

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_109,}

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 81"];
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 + 8t2−20t + 27−20t−1 + 8t−2−t−3, −q5 + 3q4−7q3 + 11q2−13q + 15−13q−1 + 11q−2−7q−3 + 3q−4−q−5 }
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]= {10_109,}

[edit] Vassiliev invariants

V2 and V3: (3, 0)
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 0 72 110 18 0 0 0 0 288 0 1320 216 \frac{15071}{10} -\frac{1466}{15} \frac{10942}{15} \frac{289}{6} \frac{991}{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 = 0 is the signature of 10 81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-1012345χ
11          1-1
9         2 2
7        51 -4
5       62  4
3      75   -2
1     86    2
-1    68     2
-3   57      -2
-5  26       4
-7 15        -4
-9 2         2
-111          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −5 {\mathbb Z}
r = −4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r = 1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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 q15−3q14 + 2q13 + 9q12−21q11 + 4q10 + 43q9−60q8−10q7 + 108q6−98q5−48q4 + 172q3−109q2−89q + 199−89q−1−109q−2 + 172q−3−48q−4−98q−5 + 108q−6−10q−7−60q−8 + 43q−9 + 4q−10−21q−11 + 9q−12 + 2q−13−3q−14 + q−15
3 −q30 + 3q29−2q28−4q27 + q26 + 17q25−5q24−39q23 + 2q22 + 83q21 + 12q20−144q19−64q18 + 237q17 + 144q16−320q15−287q14 + 395q13 + 472q12−440q11−677q10 + 430q9 + 894q8−387q7−1074q6 + 290q5 + 1235q4−196q3−1308q2 + 54q + 1359 + 54q−1−1308q−2−196q−3 + 1235q−4 + 290q−5−1074q−6−387q−7 + 894q−8 + 430q−9−677q−10−440q−11 + 472q−12 + 395q−13−287q−14−320q−15 + 144q−16 + 237q−17−64q−18−144q−19 + 12q−20 + 83q−21 + 2q−22−39q−23−5q−24 + 17q−25 + q−26−4q−27−2q−28 + 3q−29−q−30
4 q50−3q49 + 2q48 + 4q47−6q46 + 3q45−16q44 + 16q43 + 34q42−29q41−14q40−87q39 + 62q38 + 185q37−25q36−96q35−399q34 + 58q33 + 615q32 + 278q31−116q30−1255q29−429q28 + 1230q27 + 1314q26 + 525q25−2569q24−1990q23 + 1284q22 + 3006q21 + 2539q20−3483q19−4520q18−33q17 + 4439q16 + 5724q15−3114q14−6965q13−2568q12 + 4730q11 + 8927q10−1545q9−8332q8−5289q7 + 3864q6 + 11073q5 + 460q4−8389q3−7331q2 + 2332q + 11797 + 2332q−1−7331q−2−8389q−3 + 460q−4 + 11073q−5 + 3864q−6−5289q−7−8332q−8−1545q−9 + 8927q−10 + 4730q−11−2568q−12−6965q−13−3114q−14 + 5724q−15 + 4439q−16−33q−17−4520q−18−3483q−19 + 2539q−20 + 3006q−21 + 1284q−22−1990q−23−2569q−24 + 525q−25 + 1314q−26 + 1230q−27−429q−28−1255q−29−116q−30 + 278q−31 + 615q−32 + 58q−33−399q−34−96q−35−25q−36 + 185q−37 + 62q−38−87q−39−14q−40−29q−41 + 34q−42 + 16q−43−16q−44 + 3q−45−6q−46 + 4q−47 + 2q−48−3q−49 + q−50
5 −q75 + 3q74−2q73−4q72 + 6q71 + 2q70−4q69 + 5q68−11q67−22q66 + 23q65 + 43q64 + 11q63−14q62−90q61−112q60 + 40q59 + 238q58 + 245q57 + 2q56−416q55−645q54−200q53 + 710q52 + 1312q51 + 783q50−897q49−2429q48−2020q47 + 699q46 + 3831q45 + 4318q44 + 432q43−5363q42−7663q41−3090q40 + 6098q39 + 12133q38 + 7872q37−5472q36−16917q35−14774q34 + 2252q33 + 21133q32 + 23676q31 + 3836q30−23638q29−33459q28−12909q27 + 23384q26 + 43029q25 + 24403q24−20125q23−51135q22−36996q21 + 13867q20 + 56767q19 + 49718q18−5493q17−59785q16−60976q15−4204q14 + 60057q13 + 70514q12 + 13932q11−58298q10−77413q9−23291q8 + 54796q7 + 82432q6 + 31414q5−50368q4−84899q3−38762q2 + 44856q + 86051 + 44856q−1−38762q−2−84899q−3−50368q−4 + 31414q−5 + 82432q−6 + 54796q−7−23291q−8−77413q−9−58298q−10 + 13932q−11 + 70514q−12 + 60057q−13−4204q−14−60976q−15−59785q−16−5493q−17 + 49718q−18 + 56767q−19 + 13867q−20−36996q−21−51135q−22−20125q−23 + 24403q−24 + 43029q−25 + 23384q−26−12909q−27−33459q−28−23638q−29 + 3836q−30 + 23676q−31 + 21133q−32 + 2252q−33−14774q−34−16917q−35−5472q−36 + 7872q−37 + 12133q−38 + 6098q−39−3090q−40−7663q−41−5363q−42 + 432q−43 + 4318q−44 + 3831q−45 + 699q−46−2020q−47−2429q−48−897q−49 + 783q−50 + 1312q−51 + 710q−52−200q−53−645q−54−416q−55 + 2q−56 + 245q−57 + 238q−58 + 40q−59−112q−60−90q−61−14q−62 + 11q−63 + 43q−64 + 23q−65−22q−66−11q−67 + 5q−68−4q−69 + 2q−70 + 6q−71−4q−72−2q−73 + 3q−74−q−75

[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_80

10_82

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