10 27

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10_26

10_28

Contents

Image:10 27.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 27's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_27's page at Knotilus!

Visit 10 27's page at the original Knot Atlas!

[edit 10 27 Quick Notes]
[edit 10 27 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X20,13,1,14 X14,5,15,6 X6,19,7,20 X18,9,19,10 X16,7,17,8 X8,17,9,18 X10,15,11,16 X2,12,3,11
Gauss code 1, -10, 2, -1, 4, -5, 7, -8, 6, -9, 10, -2, 3, -4, 9, -7, 8, -6, 5, -3
Dowker-Thistlethwaite code 4 12 14 16 18 2 20 10 8 6
Conway Notation [321112]


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

Length is 11, width is 4,

Braid index is 4

Image:10 27_ML.gif Image:10 27_AP.gif
[{3, 11}, {2, 9}, {7, 10}, {9, 12}, {11, 5}, {1, 3}, {8, 6}, {4, 7}, {5, 8}, {6, 2}, {12, 4}, {10, 1}]

[edit Notes on presentations of 10 27]


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 27"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X12,4,13,3 X20,13,1,14 X14,5,15,6 X6,19,7,20 X18,9,19,10 X16,7,17,8 X8,17,9,18 X10,15,11,16 X2,12,3,11
In[5]:= GaussCode[K]
Out[5]= 1, -10, 2, -1, 4, -5, 7, -8, 6, -9, 10, -2, 3, -4, 9, -7, 8, -6, 5, -3
In[6]:= DTCode[K]
Out[6]= 4 12 14 16 18 2 20 10 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [321112]
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,−2,3,−2,3,3})
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:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.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 27_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 11}, {2, 9}, {7, 10}, {9, 12}, {11, 5}, {1, 3}, {8, 6}, {4, 7}, {5, 8}, {6, 2}, {12, 4}, {10, 1}]
In[14]:= Draw[ap]
Image:10 27_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-10][-2]
Hyperbolic Volume 12.3841
A-Polynomial See Data:10 27/A-polynomial

[edit Notes for 10 27'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 27's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial 2t3−8t2 + 16t−19 + 16t−1−8t−2 + 2t−3
Conway polynomial 2z6 + 4z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 71, -2 }
Jones polynomial −q2 + 3q−5 + 9q−1−11q−2 + 12q−3−11q−4 + 9q−5−6q−6 + 3q−7−q−8
HOMFLY-PT polynomial (db, data sources) −z4a6−2z2a6−a6 + z6a4 + 3z4a4 + 3z2a4 + a4 + z6a2 + 3z4a2 + 3z2a2 + a2−z4−2z2
Kauffman polynomial (db, data sources) z5a9−2z3a9 + za9 + 3z6a8−6z4a8 + 3z2a8 + 4z7a7−6z5a7 + z3a7 + 3z8a6−z6a6−3z4a6−z2a6 + a6 + z9a5 + 6z7a5−12z5a5 + 7z3a5−2za5 + 6z8a4−9z6a4 + 7z4a4−4z2a4 + a4 + z9a3 + 6z7a3−14z5a3 + 11z3a3−2za3 + 3z8a2−2z6a2−3z4a2 + 4z2a2−a2 + 4z7a−8z5a + 5z3a−za + 3z6−7z4 + 4z2 + z5a−1−2z3a−1
The A2 invariant −q24 + q22−q20−q18 + 2q16−2q14 + 2q12 + 2q6−2q4 + 3q2 + q−4−q−6
The G2 invariant q128−2q126 + 5q124−8q122 + 8q120−6q118−2q116 + 17q114−31q112 + 44q110−45q108 + 26q106 + 5q104−49q102 + 90q100−111q98 + 98q96−52q94−22q92 + 93q90−138q88 + 142q86−96q84 + 21q82 + 56q80−107q78 + 104q76−55q74−16q72 + 76q70−96q68 + 62q66 + 14q64−99q62 + 161q60−167q58 + 109q56−6q54−110q52 + 193q50−214q48 + 172q46−73q44−38q42 + 126q40−160q38 + 133q36−60q34−24q32 + 80q30−87q28 + 47q26 + 25q24−86q22 + 116q20−96q18 + 31q16 + 47q14−115q12 + 145q10−121q8 + 70q6−q4−59q2 + 94−96q−2 + 73q−4−36q−6−2q−8 + 27q−10−38q−12 + 36q−14−25q−16 + 14q−18−q−20−6q−22 + 6q−24−7q−26 + 4q−28−2q−30 + q−32
Further Quantum Invariants
Further quantum knot invariants for 10_27.

A1 Invariants.

Weight Invariant
1 −q17 + 2q15−3q13 + 3q11−2q9 + q7 + q5−2q3 + 4q−2q−1 + 2q−3−q−5
2 q48−2q46−q44 + 7q42−6q40−8q38 + 18q36−5q34−19q32 + 23q30 + 2q28−24q26 + 15q24 + 9q22−16q20−q18 + 11q16 + 2q14−17q12 + 7q10 + 19q8−23q6−q4 + 25q2−15−6q−2 + 16q−4−6q−6−6q−8 + 6q−10−q−12−2q−14 + q−16
3 −q93 + 2q91 + q89−3q87−4q85 + 6q83 + 11q81−11q79−22q77 + 12q75 + 39q73−7q71−62q69−4q67 + 84q65 + 26q63−98q61−58q59 + 104q57 + 87q55−95q53−112q51 + 74q49 + 126q47−45q45−126q43 + 14q41 + 112q39 + 18q37−88q35−48q33 + 56q31 + 77q29−22q27−98q25−17q23 + 111q21 + 52q19−113q17−87q15 + 106q13 + 107q11−80q9−117q7 + 51q5 + 117q3−23q−95q−1−q−3 + 76q−5 + 13q−7−49q−9−19q−11 + 30q−13 + 15q−15−16q−17−12q−19 + 8q−21 + 6q−23−3q−25−3q−27 + q−29 + 2q−31−q−33
4 q152−2q150−q148 + 3q146 + 4q142−9q140−6q138 + 12q136 + 6q134 + 17q132−33q130−36q128 + 25q126 + 41q124 + 72q122−71q120−130q118−13q116 + 103q114 + 240q112−42q110−288q108−204q106 + 82q104 + 505q102 + 186q100−348q98−529q96−174q94 + 662q92 + 566q90−145q88−743q86−572q84 + 520q82 + 819q80 + 232q78−654q76−828q74 + 169q72 + 759q70 + 510q68−342q66−780q64−172q62 + 465q60 + 584q58 + 13q56−528q54−418q52 + 103q50 + 531q48 + 344q46−202q44−609q42−272q40 + 412q38 + 637q36 + 167q34−698q32−627q30 + 170q28 + 796q26 + 559q24−566q22−826q20−182q18 + 667q16 + 798q14−209q12−711q10−459q8 + 296q6 + 723q4 + 121q2−360−459q−2−35q−4 + 416q−6 + 208q−8−54q−10−266q−12−138q−14 + 146q−16 + 118q−18 + 53q−20−93q−22−90q−24 + 32q−26 + 33q−28 + 40q−30−21q−32−33q−34 + 8q−36 + 3q−38 + 14q−40−3q−42−9q−44 + 3q−46 + 3q−50−q−52−2q−54 + q−56
5 −q225 + 2q223 + q221−3q219−q213 + 4q211 + 5q209−8q207−9q205 + 2q203 + 8q201 + 18q199 + 11q197−22q195−52q193−27q191 + 46q189 + 100q187 + 81q185−47q183−197q181−209q179 + 14q177 + 323q175 + 420q173 + 134q171−413q169−756q167−468q165 + 403q163 + 1157q161 + 1012q159−140q157−1494q155−1785q153−463q151 + 1622q149 + 2652q147 + 1415q145−1357q143−3394q141−2665q139 + 621q137 + 3830q135 + 3964q133 + 516q131−3746q129−5056q127−1930q125 + 3156q123 + 5727q121 + 3304q119−2140q117−5816q115−4414q113 + 895q111 + 5379q109 + 5066q107 + 328q105−4528q103−5210q101−1351q99 + 3432q97 + 4936q95 + 2093q93−2306q91−4367q89−2550q87 + 1242q85 + 3658q83 + 2838q81−284q79−2964q77−3032q75−594q73 + 2291q71 + 3251q69 + 1472q67−1657q65−3511q63−2387q61 + 967q59 + 3766q57 + 3378q55−154q53−3890q51−4390q49−844q47 + 3782q45 + 5257q43 + 1987q41−3270q39−5835q37−3212q35 + 2406q33 + 5936q31 + 4234q29−1168q27−5475q25−4935q23−182q21 + 4512q19 + 5098q17 + 1384q15−3165q13−4701q11−2265q9 + 1751q7 + 3884q5 + 2613q3−514q−2766q−1−2529q−3−364q−5 + 1708q−7 + 2072q−9 + 809q−11−805q−13−1487q−15−902q−17 + 225q−19 + 909q−21 + 769q−23 + 92q−25−486q−27−543q−29−187q−31 + 195q−33 + 335q−35 + 184q−37−66q−39−181q−41−121q−43 + 3q−45 + 81q−47 + 78q−49 + 11q−51−40q−53−36q−55−5q−57 + 13q−59 + 14q−61 + 8q−63−7q−65−10q−67 + 2q−69 + 4q−71−3q−79 + q−81 + 2q−83−q−85

A2 Invariants.

Weight Invariant
1,0 −q24 + q22−q20−q18 + 2q16−2q14 + 2q12 + 2q6−2q4 + 3q2 + q−4−q−6
1,1 q68−4q66 + 12q64−28q62 + 56q60−102q58 + 168q56−252q54 + 351q52−462q50 + 560q48−622q46 + 639q44−592q42 + 466q40−252q38−18q36 + 318q34−636q32 + 920q30−1141q28 + 1266q26−1286q24 + 1198q22−1011q20 + 748q18−446q16 + 132q14 + 156q12−374q10 + 534q8−612q6 + 626q4−570q2 + 488−394q−2 + 296q−4−204q−6 + 132q−8−84q−10 + 46q−12−22q−14 + 10q−16−4q−18 + q−20
2,0 q62−q60−q58 + 2q56 + q54−3q52−3q50 + 5q48 + 3q46−8q44−q42 + 10q40−10q36 + 2q34 + 10q32−5q30−9q28 + 6q26 + 2q24−10q22 + 2q20 + 7q18−6q16−2q14 + 11q12 + 3q10−9q8 + 3q6 + 13q4−4q2−7 + 6q−2 + 5q−4−6q−6−4q−8 + 3q−10 + 2q−12−2q−14−q−16 + q−18

A3 Invariants.

Weight Invariant
0,1,0 q54−2q52 + q50 + 4q48−8q46 + 3q44 + 10q42−16q40 + 4q38 + 15q36−21q34 + 16q30−14q28−4q26 + 11q24−5q20−q18 + 13q16−3q14−13q12 + 19q10−19q6 + 17q4 + 3q2−13 + 10q−2 + 3q−4−7q−6 + 3q−8−2q−12 + q−14
1,0,0 −q31 + q29−2q27 + q25−2q23 + 2q21−2q19 + 2q17 + q13 + q11 + 2q7−2q5 + 3q3−q + 2q−1−q−3 + q−5−q−7

A4 Invariants.

Weight Invariant
0,1,0,0 q68−q66−q64 + 4q62−6q58 + 4q56 + 8q54−7q52−8q50 + 9q48 + 5q46−16q44−6q42 + 13q40−5q38−15q36 + 10q34 + 9q32−11q30 + 3q28 + 15q26−3q24−9q22 + 12q20 + 8q18−14q16−2q14 + 15q12−2q10−12q8 + 7q6 + 8q4−3q2−3 + 5q−2 + 2q−4−3q−6−q−8 + q−10−q−12−q−14 + q−16
1,0,0,0 −q38 + q36−2q34−2q28 + 2q26−2q24 + 2q22 + q18 + q16 + q14 + q12 + 2q8−2q6 + 3q4−q2 + 1 + q−2−q−4 + q−6−q−8

B2 Invariants.

Weight Invariant
0,1 −q54 + 2q52−5q50 + 8q48−12q46 + 17q44−20q42 + 22q40−22q38 + 19q36−13q34 + 4q32 + 6q30−18q28 + 28q26−37q24 + 42q22−43q20 + 41q18−33q16 + 25q14−13q12 + 3q10 + 8q8−15q6 + 21q4−21q2 + 21−18q−2 + 15q−4−11q−6 + 7q−8−4q−10 + 2q−12−q−14
1,0 q88−2q84−2q82 + 3q80 + 6q78−q76−10q74−6q72 + 11q70 + 15q68−5q66−21q64−7q62 + 20q60 + 18q58−12q56−24q54−q52 + 21q50 + 9q48−16q46−13q44 + 10q42 + 14q40−6q38−14q36 + 3q34 + 15q32 + q30−15q28−3q26 + 16q24 + 9q22−15q20−14q18 + 12q16 + 22q14−3q12−24q10−9q8 + 20q6 + 18q4−7q2−19−3q−2 + 14q−4 + 10q−6−5q−8−9q−10−q−12 + 5q−14 + 2q−16−2q−18−2q−20 + q−24

D4 Invariants.

Weight Invariant
1,0,0,0 q74−2q72 + 3q70−4q68 + 7q66−10q64 + 11q62−13q60 + 17q58−18q56 + 16q54−16q52 + 16q50−13q48 + 3q46−3q44−3q42 + 10q40−20q38 + 21q36−25q34 + 34q32−32q30 + 32q28−31q26 + 33q24−22q22 + 19q20−15q18 + 9q16 + 2q14−4q12 + 7q10−14q8 + 18q6−15q4 + 16q2−16 + 16q−2−11q−4 + 10q−6−9q−8 + 6q−10−4q−12 + 2q−14−2q−16 + q−18

G2 Invariants.

Weight Invariant
1,0 q128−2q126 + 5q124−8q122 + 8q120−6q118−2q116 + 17q114−31q112 + 44q110−45q108 + 26q106 + 5q104−49q102 + 90q100−111q98 + 98q96−52q94−22q92 + 93q90−138q88 + 142q86−96q84 + 21q82 + 56q80−107q78 + 104q76−55q74−16q72 + 76q70−96q68 + 62q66 + 14q64−99q62 + 161q60−167q58 + 109q56−6q54−110q52 + 193q50−214q48 + 172q46−73q44−38q42 + 126q40−160q38 + 133q36−60q34−24q32 + 80q30−87q28 + 47q26 + 25q24−86q22 + 116q20−96q18 + 31q16 + 47q14−115q12 + 145q10−121q8 + 70q6−q4−59q2 + 94−96q−2 + 73q−4−36q−6−2q−8 + 27q−10−38q−12 + 36q−14−25q−16 + 14q−18−q−20−6q−22 + 6q−24−7q−26 + 4q−28−2q−30 + q−32

.

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 27"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 2t3−8t2 + 16t−19 + 16t−1−8t−2 + 2t−3
In[5]:= Conway[K][z]
Out[5]= 2z6 + 4z4 + 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]= { 71, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q2 + 3q−5 + 9q−1−11q−2 + 12q−3−11q−4 + 9q−5−6q−6 + 3q−7−q−8
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z4a6−2z2a6−a6 + z6a4 + 3z4a4 + 3z2a4 + a4 + z6a2 + 3z4a2 + 3z2a2 + a2−z4−2z2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z5a9−2z3a9 + za9 + 3z6a8−6z4a8 + 3z2a8 + 4z7a7−6z5a7 + z3a7 + 3z8a6−z6a6−3z4a6−z2a6 + a6 + z9a5 + 6z7a5−12z5a5 + 7z3a5−2za5 + 6z8a4−9z6a4 + 7z4a4−4z2a4 + a4 + z9a3 + 6z7a3−14z5a3 + 11z3a3−2za3 + 3z8a2−2z6a2−3z4a2 + 4z2a2−a2 + 4z7a−8z5a + 5z3a−za + 3z6−7z4 + 4z2 + z5a−1−2z3a−1

[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 27"];
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]= { 2t3−8t2 + 16t−19 + 16t−1−8t−2 + 2t−3, −q2 + 3q−5 + 9q−1−11q−2 + 12q−3−11q−4 + 9q−5−6q−6 + 3q−7−q−8 }
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, -3)
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 −24 32 \frac{172}{3} -\frac{52}{3} −192 −208 −32 72 \frac{256}{3} 288 \frac{1376}{3} -\frac{416}{3} \frac{12631}{15} \frac{7516}{15} -\frac{21596}{45} -\frac{343}{9} -\frac{2009}{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 27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-10123χ
5          1-1
3         2 2
1        31 -2
-1       62  4
-3      64   -2
-5     65    1
-7    56     1
-9   46      -2
-11  25       3
-13 14        -3
-15 2         2
-171          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −3 i = −1
r = −7 {\mathbb Z}
r = −6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = −1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r = 1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 3 {\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 q7−3q6 + q5 + 8q4−15q3 + q2 + 30q−37−8q−1 + 70q−2−63q−3−30q−4 + 112q−5−75q−6−54q−7 + 131q−8−66q−9−66q−10 + 116q−11−41q−12−60q−13 + 77q−14−15q−15−39q−16 + 35q−17−q−18−16q−19 + 9q−20 + q−21−3q−22 + q−23
3 −q15 + 3q14−q13−4q12−q11 + 12q10 + q9−24q8−5q7 + 43q6 + 16q5−73q4−35q3 + 105q2 + 79q−150−129q−1 + 177q−2 + 219q−3−216q−4−297q−5 + 214q−6 + 406q−7−217q−8−490q−9 + 188q−10 + 571q−11−158q−12−618q−13 + 107q−14 + 647q−15−59q−16−639q−17 + 3q−18 + 607q−19 + 47q−20−545q−21−95q−22 + 467q−23 + 128q−24−374q−25−147q−26 + 281q−27 + 145q−28−192q−29−130q−30 + 119q−31 + 105q−32−68q−33−72q−34 + 31q−35 + 47q−36−13q−37−26q−38 + 4q−39 + 13q−40−2q−41−4q−42−q−43 + 3q−44−q−45
4 q26−3q25 + q24 + 4q23−3q22 + 4q21−15q20 + 7q19 + 21q18−14q17 + 9q16−56q15 + 19q14 + 82q13−21q12 + 8q11−178q10 + 16q9 + 228q8 + 44q7 + 36q6−462q5−112q4 + 440q3 + 306q2 + 244q−913−536q−1 + 539q−2 + 787q−3 + 846q−4−1340q−5−1291q−6 + 287q−7 + 1289q−8 + 1853q−9−1471q−10−2140q−11−357q−12 + 1549q−13 + 2978q−14−1234q−15−2766q−16−1154q−17 + 1478q−18 + 3843q−19−764q−20−2991q−21−1838q−22 + 1141q−23 + 4250q−24−218q−25−2804q−26−2266q−27 + 620q−28 + 4140q−29 + 323q−30−2233q−31−2385q−32−17q−33 + 3532q−34 + 761q−35−1381q−36−2136q−37−607q−38 + 2535q−39 + 935q−40−495q−41−1549q−42−906q−43 + 1443q−44 + 764q−45 + 103q−46−838q−47−810q−48 + 607q−49 + 409q−50 + 284q−51−304q−52−491q−53 + 184q−54 + 123q−55 + 200q−56−58q−57−209q−58 + 47q−59 + 7q−60 + 83q−61 + q−62−66q−63 + 16q−64−9q−65 + 22q−66 + 4q−67−16q−68 + 5q−69−3q−70 + 4q−71 + q−72−3q−73 + q−74

[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_26

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