10 14

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10_13

10_15

Contents

Image:10 14.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 14's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_14's page at Knotilus!

Visit 10 14's page at the original Knot Atlas!

[edit 10 14 Quick Notes]
[edit 10 14 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,16,8,17 X13,20,14,1 X19,14,20,15 X9,18,10,19 X15,6,16,7 X17,8,18,9
Gauss code -1, 4, -3, 1, -2, 9, -5, 10, -8, 3, -4, 2, -6, 7, -9, 5, -10, 8, -7, 6
Dowker-Thistlethwaite code 4 10 12 16 18 2 20 6 8 14
Conway Notation [42112]


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

Length is 11, width is 4,

Braid index is 4

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

[edit Notes on presentations of 10 14]


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

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [42112]
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,−1,−2,1,−2,3,−2,3})
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:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.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 14_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{12, 4}, {3, 10}, {11, 5}, {4, 6}, {10, 12}, {5, 2}, {1, 3}, {2, 7}, {6, 8}, {7, 9}, {8, 11}, {9, 1}]
In[14]:= Draw[ap]
Image:10 14_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 [-13][1]
Hyperbolic Volume 10.9377
A-Polynomial See Data:10 14/A-polynomial

[edit Notes for 10 14'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 10 14's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −2t3 + 8t2−12t + 13−12t−1 + 8t−2−2t−3
Conway polynomial −2z6−4z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 57, -4 }
Jones polynomial 1−2q−1 + 4q−2−6q−3 + 9q−4−9q−5 + 9q−6−8q−7 + 5q−8−3q−9 + q−10
HOMFLY-PT polynomial (db, data sources) z4a8 + 2z2a8−z6a6−3z4a6−2z2a6−a6−z6a4−3z4a4−z2a4 + a4 + z4a2 + 3z2a2 + a2
Kauffman polynomial (db, data sources) z4a12−z2a12 + 3z5a11−4z3a11 + za11 + 4z6a10−4z4a10 + 4z7a9−4z5a9 + 2za9 + 3z8a8−4z6a8 + 5z4a8−3z2a8 + z9a7 + 3z7a7−9z5a7 + 8z3a7−2za7 + 5z8a6−14z6a6 + 16z4a6−9z2a6 + a6 + z9a5 + z7a5−9z5a5 + 10z3a5−4za5 + 2z8a4−5z6a4 + 2z4a4−z2a4 + a4 + 2z7a3−7z5a3 + 6z3a3−za3 + z6a2−4z4a2 + 4z2a2−a2
The A2 invariant q30−q28−2q22 + q20−2q18 + q16 + q14 + 3q10−q8 + q6 + 1
The G2 invariant q162−2q160 + 4q158−6q156 + 4q154−2q152−4q150 + 12q148−18q146 + 22q144−20q142 + 10q140 + 5q138−22q136 + 40q134−46q132 + 41q130−27q128 + 29q124−52q122 + 63q120−50q118 + 25q116 + 10q114−37q112 + 44q110−33q108 + 8q106 + 21q104−42q102 + 36q100−7q98−31q96 + 66q94−81q92 + 59q90−21q88−33q86 + 73q84−97q82 + 88q80−51q78 + q76 + 45q74−72q72 + 69q70−43q68 + 6q66 + 27q64−42q62 + 39q60−7q58−22q56 + 51q54−54q52 + 31q50 + 5q48−43q46 + 67q44−66q42 + 47q40−12q38−24q36 + 48q34−53q32 + 43q30−23q28 + q26 + 14q24−21q22 + 22q20−15q18 + 9q16−3q12 + 4q10−4q8 + 3q6−q4 + q2
Further Quantum Invariants
Further quantum knot invariants for 10_14.

A1 Invariants.

Weight Invariant
1 q21−2q19 + 2q17−3q15 + q13 + 3q7−2q5 + 2q3−q + q−1
2 q58−2q56−q54 + 5q52−5q50−2q48 + 12q46−8q44−7q42 + 16q40−6q38−10q36 + 10q34−8q30−q28 + 8q26−q24−10q22 + 10q20 + 6q18−15q16 + 6q14 + 10q12−11q10 + q8 + 9q6−5q4−2q2 + 4−q−2−q−4 + q−6
3 q111−2q109−q107 + 2q105 + 3q103−2q101−5q99 + 6q97 + 4q95−9q93−7q91 + 16q89 + 11q87−25q85−22q83 + 33q81 + 32q79−33q77−45q75 + 31q73 + 55q71−20q69−54q67 + 5q65 + 51q63 + 8q61−36q59−26q57 + 23q55 + 32q53−7q51−43q49−8q47 + 45q45 + 22q43−47q41−32q39 + 43q37 + 43q35−33q33−54q31 + 22q29 + 55q27−4q25−52q23−9q21 + 44q19 + 21q17−30q15−25q13 + 17q11 + 24q9−6q7−17q5 + 12q + 2q−1−6q−3−2q−5 + 3q−7 + q−9−q−11−q−13 + q−15
4 q180−2q178−q176 + 2q174 + 6q170−5q168−4q166 + q164−6q162 + 15q160−5q158 + q156 + 7q154−24q152 + 5q150−14q148 + 26q146 + 50q144−26q142−35q140−82q138 + 30q136 + 141q134 + 44q132−61q130−208q128−45q126 + 204q124 + 178q122 + 9q120−289q118−185q116 + 147q114 + 259q112 + 140q110−223q108−255q106−q104 + 196q102 + 216q100−54q98−195q96−124q94 + 51q92 + 188q90 + 94q88−78q86−182q84−71q82 + 126q80 + 188q78 + 14q76−207q74−151q72 + 70q70 + 260q68 + 93q66−213q64−221q62−9q60 + 288q58 + 187q56−141q54−253q52−141q50 + 213q48 + 254q46 + 10q44−184q42−237q40 + 50q38 + 204q36 + 135q34−29q32−211q30−84q28 + 66q26 + 137q24 + 88q22−92q20−95q18−38q16 + 54q14 + 92q12−4q10−36q8−48q6−4q4 + 42q2 + 12−20q−4−10q−6 + 12q−8 + 3q−10 + 4q−12−4q−14−4q−16 + 3q−18 + q−22−q−24−q−26 + q−28
5 q265−2q263−q261 + 2q259 + 3q255 + 3q253−4q251−9q249−2q247 + q245 + 8q243 + 16q241 + 6q239−13q237−27q235−19q233 + q231 + 33q229 + 59q227 + 30q225−40q223−93q221−87q219−4q217 + 132q215 + 197q213 + 75q211−147q209−315q207−240q205 + 102q203 + 458q201 + 474q199 + 29q197−562q195−766q193−275q191 + 583q189 + 1062q187 + 637q185−469q183−1304q181−1034q179 + 207q177 + 1382q175 + 1416q173 + 181q171−1290q169−1665q167−593q165 + 999q163 + 1711q161 + 952q159−581q157−1560q155−1178q153 + 151q151 + 1217q149 + 1216q147 + 261q145−813q143−1122q141−523q139 + 392q137 + 916q135 + 710q133−41q131−714q129−778q127−225q125 + 525q123 + 849q121 + 411q119−410q117−903q115−575q113 + 339q111 + 1006q109 + 728q107−286q105−1107q103−926q101 + 186q99 + 1212q97 + 1142q95−27q93−1236q91−1350q89−235q87 + 1143q85 + 1524q83 + 541q81−920q79−1563q77−862q75 + 544q73 + 1465q71 + 1128q69−126q67−1190q65−1236q63−315q61 + 784q59 + 1187q57 + 640q55−328q53−947q51−805q49−98q47 + 600q45 + 788q43 + 392q41−229q39−611q37−515q35−77q33 + 359q31 + 489q29 + 258q27−121q25−353q23−303q21−54q19 + 194q17 + 263q15 + 129q13−64q11−171q9−135q7−13q5 + 90q3 + 102q + 40q−1−34q−3−62q−5−34q−7 + 5q−9 + 28q−11 + 26q−13 + 3q−15−14q−17−11q−19−2q−21 + q−23 + 6q−25 + 4q−27−3q−29−2q−31 + q−33 + q−39−q−41−q−43 + q−45

A2 Invariants.

Weight Invariant
1,0 q30−q28−2q22 + q20−2q18 + q16 + q14 + 3q10−q8 + q6 + 1
1,1 q84−4q82 + 10q80−20q78 + 34q76−54q74 + 78q72−106q70 + 136q68−164q66 + 190q64−208q62 + 217q60−202q58 + 164q56−106q54 + 23q52 + 80q50−192q48 + 298q46−388q44 + 448q42−480q40 + 470q38−426q36 + 346q34−244q32 + 130q30−17q28−88q26 + 174q24−226q22 + 252q20−246q18 + 226q16−186q14 + 144q12−102q10 + 70q8−42q6 + 25q4−12q2 + 6−2q−2 + q−4
2,0 q76−q74−q72 + 3q62−q60−2q58 + 4q56 + 5q54−5q52−3q50 + 5q48−9q44−4q42 + 5q40−3q38−4q36 + 5q34 + 3q32−4q30 + 3q28 + 4q26−3q24−2q22 + 6q20 + 2q18−6q16 + q14 + 7q12−4q8 + q6 + 3q4−1 + q−4

A3 Invariants.

Weight Invariant
0,1,0 q68−2q66 + 4q62−6q60−q58 + 9q56−8q54−3q52 + 15q50−6q48−5q46 + 10q44−4q42−7q40 + q38 + 2q36−3q34−6q32 + 6q30 + 3q28−12q26 + 7q24 + 7q22−10q20 + 6q18 + 7q16−6q14 + 4q12 + 3q10−3q8 + 2q6 + q4−q2 + 1
1,0,0 q39−q37 + q35−2q33 + q31−2q29 + q27−2q25 + 2q17 + 3q13−q11 + 2q9−q7 + q5 + q

A4 Invariants.

Weight Invariant
0,1,0,0 q86−q84−2q82 + 3q80 + 2q78−6q76−2q74 + 6q72−q70−7q68 + 3q66 + 10q64−2q62−3q60 + 9q58 + 2q56−9q54 + 2q52 + 3q50−10q48−5q46 + 5q44−4q42−11q40 + 2q38 + 7q36−5q34−4q32 + 9q30 + 4q28−2q26 + 3q24 + 5q22 + q20 + q18 + 2q16 + q14 + q10 + q8 + q2
1,0,0,0 q48−q46 + q44−q42−q40 + q38−2q36 + q34−2q32−q28 + q22 + 2q20 + 3q16−q14 + 2q12 + q6 + q2

B2 Invariants.

Weight Invariant
0,1 q68−2q66 + 4q64−6q62 + 8q60−11q58 + 13q56−14q54 + 13q52−11q50 + 6q48−q46−6q44 + 12q42−19q40 + 23q38−26q36 + 27q34−24q32 + 20q30−13q28 + 8q26−q24−5q22 + 10q20−12q18 + 13q16−12q14 + 12q12−9q10 + 7q8−4q6 + 3q4−q2 + 1
1,0 q110−2q106−2q104 + 2q102 + 5q100−7q96−6q94 + 5q92 + 11q90 + q88−12q86−8q84 + 9q82 + 15q80−q78−13q76−4q74 + 11q72 + 7q70−8q68−10q66 + 3q64 + 8q62−2q60−10q58−2q56 + 8q54 + 2q52−8q50−4q48 + 9q46 + 7q44−7q42−12q40 + 4q38 + 15q36 + 4q34−13q32−10q30 + 8q28 + 14q26 + q24−10q22−5q20 + 7q18 + 7q16−q14−5q12−q10 + 3q8 + 2q6−q4−q2 + q−2

D4 Invariants.

Weight Invariant
1,0,0,0 q94−2q92 + 2q90−3q88 + 5q86−7q84 + 6q82−8q80 + 11q78−11q76 + 9q74−9q72 + 12q70−5q68 + 3q66−q64−2q62 + 8q60−12q58 + 11q56−18q54 + 19q52−20q50 + 19q48−22q46 + 17q44−15q42 + 11q40−11q38 + 4q36−q32 + 6q30−6q28 + 12q26−8q24 + 11q22−9q20 + 11q18−7q16 + 7q14−5q12 + 5q10−2q8 + 2q6−q4 + q2

G2 Invariants.

Weight Invariant
1,0 q162−2q160 + 4q158−6q156 + 4q154−2q152−4q150 + 12q148−18q146 + 22q144−20q142 + 10q140 + 5q138−22q136 + 40q134−46q132 + 41q130−27q128 + 29q124−52q122 + 63q120−50q118 + 25q116 + 10q114−37q112 + 44q110−33q108 + 8q106 + 21q104−42q102 + 36q100−7q98−31q96 + 66q94−81q92 + 59q90−21q88−33q86 + 73q84−97q82 + 88q80−51q78 + q76 + 45q74−72q72 + 69q70−43q68 + 6q66 + 27q64−42q62 + 39q60−7q58−22q56 + 51q54−54q52 + 31q50 + 5q48−43q46 + 67q44−66q42 + 47q40−12q38−24q36 + 48q34−53q32 + 43q30−23q28 + q26 + 14q24−21q22 + 22q20−15q18 + 9q16−3q12 + 4q10−4q8 + 3q6−q4 + q2

.

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 14"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −2t3 + 8t2−12t + 13−12t−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]= { 57, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= 1−2q−1 + 4q−2−6q−3 + 9q−4−9q−5 + 9q−6−8q−7 + 5q−8−3q−9 + q−10
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z4a8 + 2z2a8−z6a6−3z4a6−2z2a6−a6−z6a4−3z4a4−z2a4 + a4 + z4a2 + 3z2a2 + a2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z4a12−z2a12 + 3z5a11−4z3a11 + za11 + 4z6a10−4z4a10 + 4z7a9−4z5a9 + 2za9 + 3z8a8−4z6a8 + 5z4a8−3z2a8 + z9a7 + 3z7a7−9z5a7 + 8z3a7−2za7 + 5z8a6−14z6a6 + 16z4a6−9z2a6 + a6 + z9a5 + z7a5−9z5a5 + 10z3a5−4za5 + 2z8a4−5z6a4 + 2z4a4−z2a4 + a4 + 2z7a3−7z5a3 + 6z3a3−za3 + z6a2−4z4a2 + 4z2a2−a2

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

Same Alexander/Conway Polynomial: {K11a161, K11n2,}

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 14"];
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 + 8t2−12t + 13−12t−1 + 8t−2−2t−3, 1−2q−1 + 4q−2−6q−3 + 9q−4−9q−5 + 9q−6−8q−7 + 5q−8−3q−9 + q−10 }
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]= {K11a161, K11n2,}
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{364}{3} \frac{140}{3} −192 −592 −160 −184 \frac{256}{3} 288 \frac{2912}{3} \frac{1120}{3} \frac{39031}{15} -\frac{5764}{15} \frac{92644}{45} \frac{617}{9} \frac{4711}{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 = -4 is the signature of 10 14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -8-7-6-5-4-3-2-1012χ
1          11
-1         1 -1
-3        31 2
-5       42  -2
-7      52   3
-9     44    0
-11    55     0
-13   34      1
-15  25       -3
-17 13        2
-19 2         -2
-211          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −5 i = −3
r = −8 {\mathbb Z}
r = −7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r = 1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 2 {\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 q2−2q + 6q−1−8q−2−3q−3 + 20q−4−16q−5−15q−6 + 41q−7−20q−8−36q−9 + 62q−10−16q−11−56q−12 + 71q−13−7q−14−65q−15 + 64q−16 + q−17−55q−18 + 44q−19 + 5q−20−33q−21 + 21q−22 + 4q−23−13q−24 + 7q−25 + q−26−3q−27 + q−28
3 q6−2q5 + 2q3 + 3q2−7q−4 + 10q−1 + 13q−2−19q−3−21q−4 + 21q−5 + 43q−6−26q−7−63q−8 + 16q−9 + 94q−10−3q−11−116q−12−27q−13 + 142q−14 + 56q−15−149q−16−103q−17 + 163q−18 + 132q−19−149q−20−178q−21 + 148q−22 + 201q−23−126q−24−231q−25 + 113q−26 + 237q−27−87q−28−240q−29 + 64q−30 + 227q−31−43q−32−197q−33 + 18q−34 + 168q−35−9q−36−122q−37−6q−38 + 92q−39 + 3q−40−57q−41−5q−42 + 37q−43−21q−45 + 14q−47−2q−48−8q−49 + 2q−50 + 3q−51 + q−52−3q−53 + q−54
4 q12−2q11 + 2q9−q8 + 4q7−9q6 + 10q4−2q3 + 13q2−31q−10 + 30q−1 + 10q−2 + 43q−3−77q−4−54q−5 + 42q−6 + 42q−7 + 139q−8−115q−9−146q−10−15q−11 + 45q−12 + 319q−13−66q−14−217q−15−165q−16−82q−17 + 501q−18 + 98q−19−148q−20−319q−21−369q−22 + 554q−23 + 292q−24 + 96q−25−360q−26−723q−27 + 442q−28 + 404q−29 + 424q−30−259q−31−1020q−32 + 230q−33 + 412q−34 + 730q−35−92q−36−1210q−37 + 9q−38 + 356q−39 + 951q−40 + 82q−41−1272q−42−188q−43 + 245q−44 + 1055q−45 + 254q−46−1178q−47−325q−48 + 70q−49 + 984q−50 + 395q−51−908q−52−345q−53−127q−54 + 730q−55 + 427q−56−545q−57−226q−58−239q−59 + 398q−60 + 323q−61−247q−62−57q−63−213q−64 + 149q−65 + 160q−66−100q−67 + 48q−68−116q−69 + 38q−70 + 48q−71−53q−72 + 57q−73−40q−74 + 14q−75 + 8q−76−34q−77 + 28q−78−9q−79 + 8q−80 + 2q−81−14q−82 + 7q−83−2q−84 + 3q−85 + q−86−3q−87 + q−88

[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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