10 22

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10_21

10_23

Contents

Image:10 22.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 22's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_22's page at Knotilus!

Visit 10 22's page at the original Knot Atlas!

[edit 10 22 Quick Notes]
[edit 10 22 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X18,8,19,7 X20,10,1,9 X8,20,9,19 X4,13,5,14 X10,18,11,17
Gauss code 1, -4, 3, -9, 5, -1, 6, -8, 7, -10, 2, -3, 9, -5, 4, -2, 10, -6, 8, -7
Dowker-Thistlethwaite code 6 12 14 18 20 16 4 2 10 8
Conway Notation [3313]


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

Length is 11, width is 4,

Braid index is 4

Image:10 22_ML.gif Image:10 22_AP.gif
[{13, 5}, {4, 10}, {6, 11}, {5, 7}, {10, 12}, {11, 13}, {8, 6}, {7, 2}, {3, 1}, {2, 9}, {1, 8}, {9, 4}, {12, 3}]

[edit Notes on presentations of 10 22]


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 22"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X18,8,19,7 X20,10,1,9 X8,20,9,19 X4,13,5,14 X10,18,11,17
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -9, 5, -1, 6, -8, 7, -10, 2, -3, 9, -5, 4, -2, 10, -6, 8, -7
In[6]:= DTCode[K]
Out[6]= 6 12 14 18 20 16 4 2 10 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [3313]
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,{1,1,1,1,2,−1,−3,2,−3,−3,−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, 11, 4 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.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:10 22_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{13, 5}, {4, 10}, {6, 11}, {5, 7}, {10, 12}, {11, 13}, {8, 6}, {7, 2}, {3, 1}, {2, 9}, {1, 8}, {9, 4}, {12, 3}]
In[14]:= Draw[ap]
Image:10 22_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-7]
Hyperbolic Volume 9.98187
A-Polynomial See Data:10 22/A-polynomial

[edit Notes for 10 22's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for 10 22's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −2t3 + 6t2−10t + 13−10t−1 + 6t−2−2t−3
Conway polynomial −2z6−6z4−4z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 49, 0 }
Jones polynomial q6−2q5 + 4q4−6q3 + 7q2−8q + 8−6q−1 + 4q−2−2q−3 + q−4
HOMFLY-PT polynomial (db, data sources) −z6a−2−z6 + a2z4−4z4a−2 + z4a−4−4z4 + 3a2z2−5z2a−2 + 3z2a−4−5z2 + 2a2−2a−2 + 2a−4−1
Kauffman polynomial (db, data sources) z9a−1 + z9a−3 + 4z8a−2 + 2z8a−4 + 2z8 + 3az7−z7a−3 + 2z7a−5 + 3a2z6−12z6a−2−6z6a−4 + z6a−6−2z6 + 2a3z5−6az5−z5a−1−7z5a−5 + a4z4−6a2z4 + 16z4a−2 + 6z4a−4−4z4a−6−z4−3a3z3 + 7az3−4z3a−3 + 6z3a−5−2a4z2 + 6a2z2−12z2a−2−6z2a−4 + 4z2a−6 + 6z2−az + za−1 + za−3−za−5−2a2 + 2a−2 + 2a−4−1
The A2 invariant q12 + q8 + q6−q4 + 2q2−1−q−4−2q−6 + q−8−q−10 + q−12 + q−14 + q−18
The G2 invariant q66−q64 + 2q62−3q60 + 2q58−q56−2q54 + 6q52−7q50 + 10q48−10q46 + 6q44 + q42−10q40 + 18q38−22q36 + 21q34−15q32 + 4q30 + 12q28−22q26 + 33q24−29q22 + 21q20−6q18−10q16 + 24q14−27q12 + 23q10−3q8−11q6 + 19q4−18q2 + 3 + 17q−2−36q−4 + 35q−6−27q−8 + 29q−12−52q−14 + 54q−16−43q−18 + 14q−20 + 13q−22−39q−24 + 48q−26−42q−28 + 24q−30 + q−32−21q−34 + 31q−36−23q−38 + 7q−40 + 12q−42−25q−44 + 26q−46−14q−48−7q−50 + 31q−52−40q−54 + 39q−56−20q−58−5q−60 + 26q−62−36q−64 + 37q−66−25q−68 + 8q−70 + 8q−72−18q−74 + 20q−76−15q−78 + 9q−80−2q−82−3q−84 + 4q−86−5q−88 + 3q−90−q−92 + q−94
Further Quantum Invariants
Further quantum knot invariants for 10_22.

A1 Invariants.

Weight Invariant
1 q9−q7 + 2q5−2q3 + 2q−q−3 + q−5−2q−7 + 2q−9−q−11 + q−13
2 q26−q24 + 3q20−3q18−q16 + 6q14−6q12−3q10 + 10q8−7q6−5q4 + 10q2−1−4q−2 + 2q−4 + 5q−6−3q−8−5q−10 + 8q−12 + q−14−9q−16 + 6q−18 + 5q−20−10q−22 + 2q−24 + 6q−26−6q−28−q−30 + 4q−32−q−34−q−36 + q−38
3 q51−q49 + q45 + q43−2q41−q39 + 2q37 + q35−4q33−q31 + 6q29 + q27−10q25−q23 + 16q21 + 3q19−22q17−9q15 + 29q13 + 15q11−29q9−20q7 + 22q5 + 26q3−13q−24q−1 + 4q−3 + 21q−5 + 11q−7−17q−9−19q−11 + 9q−13 + 26q−15−7q−17−30q−19 + 31q−23 + 7q−25−31q−27−12q−29 + 27q−31 + 20q−33−21q−35−25q−37 + 12q−39 + 30q−41−3q−43−26q−45−6q−47 + 21q−49 + 11q−51−14q−53−12q−55 + 5q−57 + 10q−59−q−61−6q−63−q−65 + 3q−67 + q−69−q−71−q−73 + q−75
4 q84−q82 + q78−q76 + 2q74−3q72 + 2q68−4q66 + 6q64−3q62 + 2q58−10q56 + 9q54−q52 + 7q50 + 4q48−25q46−q42 + 32q40 + 29q38−42q36−40q34−30q32 + 64q30 + 94q28−22q26−85q24−105q22 + 47q20 + 154q18 + 47q16−74q14−160q12−23q10 + 130q8 + 101q6 + 3q4−127q2−81 + 40q−2 + 90q−4 + 67q−6−39q−8−85q−10−47q−12 + 41q−14 + 91q−16 + 32q−18−69q−20−95q−22 + 11q−24 + 98q−26 + 74q−28−59q−30−127q−32−9q−34 + 100q−36 + 111q−38−38q−40−143q−42−49q−44 + 67q−46 + 143q−48 + 21q−50−119q−52−95q−54−11q−56 + 126q−58 + 87q−60−36q−62−89q−64−90q−66 + 49q−68 + 94q−70 + 44q−72−24q−74−96q−76−24q−78 + 36q−80 + 56q−82 + 32q−84−44q−86−34q−88−12q−90 + 19q−92 + 33q−94−4q−96−9q−98−15q−100−3q−102 + 12q−104 + q−106 + 2q−108−4q−110−3q−112 + 3q−114 + q−118−q−120−q−122 + q−124
5 q125−q123 + q119−q117 + q113−2q111−q109 + 2q107 + 4q101−2q99−6q97−2q95 + q93 + 6q91 + 11q89 + q87−14q85−17q83−7q81 + 16q79 + 31q77 + 23q75−13q73−51q71−53q69−3q67 + 69q65 + 102q63 + 51q61−71q59−172q57−136q55 + 44q53 + 241q51 + 261q49 + 38q47−286q45−417q43−178q41 + 281q39 + 560q37 + 357q35−192q33−644q31−563q29 + 41q27 + 643q25 + 710q23 + 156q21−535q19−766q17−355q15 + 352q13 + 724q11 + 477q9−132q7−572q5−525q3−74q + 381q−1 + 490q−3 + 219q−5−177q−7−388q−9−301q−11 + 2q−13 + 293q−15 + 334q−17 + 105q−19−199q−21−345q−23−188q−25 + 157q−27 + 367q−29 + 222q−31−146q−33−395q−35−273q−37 + 148q−39 + 458q−41 + 326q−43−142q−45−516q−47−409q−49 + 103q−51 + 553q−53 + 514q−55−17q−57−552q−59−602q−61−119q−63 + 479q−65 + 663q−67 + 278q−69−336q−71−653q−73−430q−75 + 140q−77 + 567q−79 + 522q−81 + 80q−83−395q−85−539q−87−265q−89 + 186q−91 + 452q−93 + 370q−95 + 37q−97−303q−99−390q−101−189q−103 + 119q−105 + 309q−107 + 266q−109 + 44q−111−187q−113−255q−115−137q−117 + 56q−119 + 177q−121 + 165q−123 + 41q−125−91q−127−132q−129−77q−131 + 15q−133 + 76q−135 + 75q−137 + 25q−139−29q−141−48q−143−32q−145 + q−147 + 20q−149 + 22q−151 + 11q−153−6q−155−12q−157−6q−159 + q−163 + 5q−165 + 2q−167−3q−169−q−171 + q−173 + q−179−q−181−q−183 + q−185

A2 Invariants.

Weight Invariant
1,0 q12 + q8 + q6−q4 + 2q2−1−q−4−2q−6 + q−8−q−10 + q−12 + q−14 + q−18
1,1 q36−2q34 + 4q32−8q30 + 15q28−18q26 + 28q24−40q22 + 54q20−62q18 + 72q16−90q14 + 96q12−92q10 + 88q8−82q6 + 56q4−18q2−24 + 70q−2−124q−4 + 172q−6−206q−8 + 234q−10−233q−12 + 224q−14−186q−16 + 148q−18−95q−20 + 28q−22 + 24q−24−70q−26 + 102q−28−130q−30 + 134q−32−120q−34 + 104q−36−86q−38 + 62q−40−38q−42 + 25q−44−14q−46 + 6q−48−2q−50 + q−52
2,0 q32 + 2q26 + 2q24−q22 + 2q18−q16−5q14 + 3q10−5q8−4q6 + 4q4 + 2q2−2 + q−2 + 5q−4−q−6−2q−8 + 4q−10 + 2q−12−2q−14 + 3q−16 + 5q−18−2q−20−2q−22 + q−24−5q−28−3q−30 + 2q−32−2q−36 + 2q−40 + q−42 + q−48

A3 Invariants.

Weight Invariant
0,1,0 q28−q26 + 2q22−3q20 + 7q16−4q14−2q12 + 10q10−3q8−5q6 + 8q4−2q2−5 + 2q−2−3q−6−4q−8 + 4q−10 + 2q−12−7q−14 + 4q−16 + 6q−18−7q−20 + 3q−22 + 6q−24−5q−26 + 2q−28 + 2q−30−4q−32 + q−34 + q−36−q−38 + q−40
1,0,0 q15 + 2q11 + 2q7−q5 + 2q3−q−q−3−2q−5−q−7−2q−9 + q−11−q−13 + 2q−15 + 2q−19 + q−23

A4 Invariants.

Weight Invariant
0,1,0,0 q34 + q28−q24 + q22 + 4q20 + 2q18−q16 + 3q14 + 6q12−2q10−6q8 + 3q6−q4−7q2−3 + 3q−2−2q−4−4q−6 + 5q−8 + 4q−10−3q−12 + q−14 + 8q−16−4q−18−5q−20 + 5q−22 + 2q−24−5q−26−q−28 + 5q−30−3q−34 + q−36 + 2q−38−2q−40 + 2q−44 + q−50
1,0,0,0 q18 + 2q14 + q12 + q10 + 2q8−q6 + 2q4−q2−q−2−q−4−2q−6−2q−8−q−10−2q−12 + q−14−q−16 + 2q−18 + q−20 + q−22 + 2q−24 + q−28

B2 Invariants.

Weight Invariant
0,1 q28−q26 + 2q24−4q22 + 5q20−6q18 + 9q16−8q14 + 10q12−8q10 + 7q8−3q6 + 6q2−11 + 14q−2−18q−4 + 17q−6−20q−8 + 16q−10−14q−12 + 9q−14−4q−16 + 5q−20−7q−22 + 10q−24−9q−26 + 10q−28−8q−30 + 6q−32−5q−34 + 3q−36−q−38 + q−40
1,0 q46−q42−q40 + q38 + 3q36−4q32−3q30 + 3q28 + 8q26 + q24−7q22−6q20 + 5q18 + 10q16 + q14−9q12−5q10 + 6q8 + 8q6−4q4−8q2 + 1 + 7q−2 + q−4−7q−6−3q−8 + 4q−10 + 4q−12−5q−14−4q−16 + 4q−18 + 6q−20−3q−22−8q−24 + q−26 + 10q−28 + 5q−30−8q−32−9q−34 + 4q−36 + 11q−38 + 2q−40−8q−42−6q−44 + 5q−46 + 6q−48−q−50−5q−52−2q−54 + 3q−56 + 2q−58−q−60−q−62 + q−66

D4 Invariants.

Weight Invariant
1,0,0,0 q38−q36 + q34−2q32 + 3q30−4q28 + 4q26−4q24 + 8q22−6q20 + 7q18−5q16 + 10q14−4q12 + 4q10−3q8 + 2q6 + 3q4−6q2 + 6−12q−2 + 11q−4−14q−6 + 12q−8−17q−10 + 13q−12−13q−14 + 11q−16−10q−18 + 6q−20−3q−22 + 2q−24 + 2q−26−2q−28 + 8q−30−5q−32 + 8q−34−7q−36 + 9q−38−7q−40 + 5q−42−6q−44 + 4q−46−3q−48 + 2q−50−q−52 + q−54

G2 Invariants.

Weight Invariant
1,0 q66−q64 + 2q62−3q60 + 2q58−q56−2q54 + 6q52−7q50 + 10q48−10q46 + 6q44 + q42−10q40 + 18q38−22q36 + 21q34−15q32 + 4q30 + 12q28−22q26 + 33q24−29q22 + 21q20−6q18−10q16 + 24q14−27q12 + 23q10−3q8−11q6 + 19q4−18q2 + 3 + 17q−2−36q−4 + 35q−6−27q−8 + 29q−12−52q−14 + 54q−16−43q−18 + 14q−20 + 13q−22−39q−24 + 48q−26−42q−28 + 24q−30 + q−32−21q−34 + 31q−36−23q−38 + 7q−40 + 12q−42−25q−44 + 26q−46−14q−48−7q−50 + 31q−52−40q−54 + 39q−56−20q−58−5q−60 + 26q−62−36q−64 + 37q−66−25q−68 + 8q−70 + 8q−72−18q−74 + 20q−76−15q−78 + 9q−80−2q−82−3q−84 + 4q−86−5q−88 + 3q−90−q−92 + q−94

.

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 22"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −2t3 + 6t2−10t + 13−10t−1 + 6t−2−2t−3
In[5]:= Conway[K][z]
Out[5]= −2z6−6z4−4z2 + 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]= { 49, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−2q5 + 4q4−6q3 + 7q2−8q + 8−6q−1 + 4q−2−2q−3 + q−4
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z6a−2−z6 + a2z4−4z4a−2 + z4a−4−4z4 + 3a2z2−5z2a−2 + 3z2a−4−5z2 + 2a2−2a−2 + 2a−4−1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z9a−1 + z9a−3 + 4z8a−2 + 2z8a−4 + 2z8 + 3az7−z7a−3 + 2z7a−5 + 3a2z6−12z6a−2−6z6a−4 + z6a−6−2z6 + 2a3z5−6az5−z5a−1−7z5a−5 + a4z4−6a2z4 + 16z4a−2 + 6z4a−4−4z4a−6−z4−3a3z3 + 7az3−4z3a−3 + 6z3a−5−2a4z2 + 6a2z2−12z2a−2−6z2a−4 + 4z2a−6 + 6z2−az + za−1 + za−3−za−5−2a2 + 2a−2 + 2a−4−1

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_35,}

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 22"];
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]= { −2t3 + 6t2−10t + 13−10t−1 + 6t−2−2t−3, q6−2q5 + 4q4−6q3 + 7q2−8q + 8−6q−1 + 4q−2−2q−3 + q−4 }
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_35,}

[edit] Vassiliev invariants

V2 and V3: (-4, -2)
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
−16 −16 128 \frac{664}{3} \frac{344}{3} 256 \frac{1472}{3} \frac{320}{3} 144 -\frac{2048}{3} 128 -\frac{10624}{3} -\frac{5504}{3} -\frac{44342}{15} \frac{20408}{15} -\frac{179048}{45} \frac{5126}{9} -\frac{15302}{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 = 0 is the signature of 10 22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-10123456χ
13          11
11         1 -1
9        31 2
7       31  -2
5      43   1
3     43    -1
1    44     0
-1   35      2
-3  13       -2
-5 13        2
-7 1         -1
-91          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −4 {\mathbb Z}
r = −3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r = 1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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 q18−2q17 + 6q15−7q14−5q13 + 18q12−11q11−17q10 + 33q9−10q8−32q7 + 43q6−3q5−45q4 + 45q3 + 5q2−48q + 39 + 8q−1−37q−2 + 24q−3 + 6q−4−20q−5 + 11q−6 + 3q−7−8q−8 + 4q−9 + q−10−2q−11 + q−12
3 q36−2q35 + 2q33 + 3q32−6q31−5q30 + 7q29 + 14q28−11q27−22q26 + 5q25 + 39q24−q23−49q22−15q21 + 62q20 + 32q19−67q18−52q17 + 66q16 + 73q15−60q14−91q13 + 47q12 + 111q11−36q10−122q9 + 17q8 + 134q7−3q6−139q5−11q4 + 136q3 + 25q2−129q−28 + 108q−1 + 36q−2−90q−3−32q−4 + 66q−5 + 27q−6−46q−7−18q−8 + 28q−9 + 14q−10−21q−11−5q−12 + 11q−13 + 5q−14−10q−15 + 4q−17 + 2q−18−5q−19 + q−20 + q−21 + q−22−2q−23 + q−24
4 q60−2q59 + 2q57−q56 + 4q55−8q54−q53 + 8q52−2q51 + 15q50−23q49−13q48 + 14q47 + 3q46 + 52q45−37q44−44q43−8q42−7q41 + 128q40−13q39−64q38−68q37−79q36 + 200q35 + 55q34−14q33−113q32−218q31 + 201q30 + 108q29 + 109q28−74q27−355q26 + 117q25 + 84q24 + 249q23 + 48q22−431q21 + q20−10q19 + 354q18 + 197q17−442q16−108q15−128q14 + 422q13 + 330q12−418q11−195q10−234q9 + 448q8 + 431q7−359q6−245q5−322q4 + 410q3 + 477q2−253q−222−372q−1 + 289q−2 + 431q−3−123q−4−124q−5−343q−6 + 136q−7 + 294q−8−37q−9−3q−10−236q−11 + 29q−12 + 142q−13−17q−14 + 60q−15−120q−16−q−17 + 48q−18−27q−19 + 58q−20−49q−21 + 2q−22 + 15q−23−26q−24 + 33q−25−20q−26 + 5q−27 + 7q−28−16q−29 + 14q−30−8q−31 + 3q−32 + 4q−33−7q−34 + 4q−35−2q−36 + q−37 + q−38−2q−39 + q−40

[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_21

10_23

Retrieved from "http://katlas.math.toronto.edu/wiki/10_22"
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