10 71

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10_70

10_72

Contents

Image:10 71.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 71's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_71's page at Knotilus!

Visit 10 71's page at the original Knot Atlas!

[edit 10 71 Quick Notes]
[edit 10 71 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,19,10,18 X15,20,16,1 X19,16,20,17 X17,11,18,10 X7283
Gauss code -1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -3, 4, -5, 3, -7, 8, -9, 6, -8, 7
Dowker-Thistlethwaite code 4 8 12 2 18 14 6 20 10 16
Conway Notation [22,21,2+]


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

Length is 10, width is 5,

Braid index is 5

Image:10 71_ML.gif Image:10 71_AP.gif
[{12, 4}, {3, 10}, {6, 11}, {10, 12}, {5, 7}, {4, 6}, {8, 5}, {7, 2}, {1, 3}, {2, 9}, {11, 8}, {9, 1}]

[edit Notes on presentations of 10 71]


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 71"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,19,10,18 X15,20,16,1 X19,16,20,17 X17,11,18,10 X7283
In[5]:= GaussCode[K]
Out[5]= -1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -3, 4, -5, 3, -7, 8, -9, 6, -8, 7
In[6]:= DTCode[K]
Out[6]= 4 8 12 2 18 14 6 20 10 16

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [22,21,2+]
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,−3,4})
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, 10, 5 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.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 71_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{12, 4}, {3, 10}, {6, 11}, {10, 12}, {5, 7}, {4, 6}, {8, 5}, {7, 2}, {1, 3}, {2, 9}, {11, 8}, {9, 1}]
In[14]:= Draw[ap]
Image:10 71_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 13.3852
A-Polynomial See Data:10 71/A-polynomial

[edit Notes for 10 71'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 71's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t3 + 7t2−18t + 25−18t−1 + 7t−2−t−3
Conway polynomial −z6 + z4 + z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 77, 0 }
Jones polynomial −q5 + 3q4−6q3 + 10q2−12q + 13−12q−1 + 10q−2−6q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) −z6 + 2a2z4 + 2z4a−2−3z4−a4z2 + 4a2z2 + 4z2a−2−z2a−4−5z2−a4 + 3a2 + 3a−2−a−4−3
Kauffman polynomial (db, data sources) az9 + z9a−1 + 3a2z8 + 3z8a−2 + 6z8 + 4a3z7 + 8az7 + 8z7a−1 + 4z7a−3 + 3a4z6 + 2a2z6 + 2z6a−2 + 3z6a−4−2z6 + a5z5−5a3z5−15az5−15z5a−1−5z5a−3 + z5a−5−6a4z4−12a2z4−12z4a−2−6z4a−4−12z4−2a5z3 + 7az3 + 7z3a−1−2z3a−5 + 4a4z2 + 10a2z2 + 10z2a−2 + 4z2a−4 + 12z2 + a5z + a3z−az−za−1 + za−3 + za−5−a4−3a2−3a−2−a−4−3
The A2 invariant −q16 + q12−2q10 + 3q8 + q6−q4 + 2q2−3 + 2q−2−q−4 + q−6 + 3q−8−2q−10 + q−12−q−16
The G2 invariant q80−2q78 + 5q76−8q74 + 8q72−7q70−2q68 + 16q66−32q64 + 47q62−52q60 + 36q58−3q56−48q54 + 102q52−137q50 + 132q48−84q46−6q44 + 105q42−179q40 + 206q38−155q36 + 56q34 + 61q32−147q30 + 164q28−107q26 + 10q24 + 90q22−135q20 + 110q18−14q16−110q14 + 208q12−235q10 + 166q8−34q6−128q4 + 254q2−299 + 253q−2−128q−4−32q−6 + 166q−8−233q−10 + 206q−12−108q−14−12q−16 + 109q−18−135q−20 + 90q−22 + 10q−24−107q−26 + 164q−28−150q−30 + 63q−32 + 55q−34−156q−36 + 206q−38−180q−40 + 106q−42−6q−44−84q−46 + 132q−48−136q−50 + 102q−52−47q−54−4q−56 + 36q−58−51q−60 + 46q−62−32q−64 + 16q−66−2q−68−7q−70 + 8q−72−8q−74 + 5q−76−2q−78 + q−80
Further Quantum Invariants
Further quantum knot invariants for 10_71.

A1 Invariants.

Weight Invariant
1 −q11 + 2q9−3q7 + 4q5−2q3 + q + q−1−2q−3 + 4q−5−3q−7 + 2q−9−q−11
2 q32−2q30−q28 + 7q26−7q24−7q22 + 21q20−9q18−22q16 + 30q14−30q10 + 21q8 + 11q6−20q4 + q2 + 15 + q−2−20q−4 + 11q−6 + 21q−8−30q−10 + 30q−14−22q−16−9q−18 + 21q−20−7q−22−7q−24 + 7q−26−q−28−2q−30 + q−32
3 −q63 + 2q61 + q59−3q57−4q55 + 7q53 + 10q51−15q49−21q47 + 21q45 + 44q43−23q41−77q39 + 16q37 + 115q35 + 8q33−148q31−54q29 + 171q27 + 99q25−165q23−144q21 + 138q19 + 175q17−96q15−182q13 + 46q11 + 168q9 + 7q7−138q5−56q3 + 101q + 101q−1−56q−3−138q−5 + 7q−7 + 167q−9 + 46q−11−180q−13−95q−15 + 174q−17 + 137q−19−144q−21−166q−23 + 99q−25 + 172q−27−54q−29−150q−31 + 8q−33 + 119q−35 + 17q−37−79q−39−25q−41 + 44q−43 + 22q−45−21q−47−15q−49 + 10q−51 + 7q−53−4q−55−3q−57 + q−59 + 2q−61−q−63
4 q104−2q102−q100 + 3q98 + 4q94−10q92−5q90 + 16q88 + 6q86 + 11q84−45q82−35q80 + 50q78 + 61q76 + 63q74−128q72−170q70 + 41q68 + 205q66 + 292q64−168q62−477q60−208q58 + 306q56 + 779q54 + 104q52−753q50−794q48 + 37q46 + 1242q44 + 765q42−599q40−1349q38−638q36 + 1194q34 + 1370q32 + 24q30−1377q28−1238q26 + 621q24 + 1436q22 + 642q20−871q18−1332q16−66q14 + 999q12 + 932q10−205q8−1027q6−598q4 + 403q2 + 976 + 409q−2−588q−4−1021q−6−209q−8 + 915q−10 + 983q−12−62q−14−1311q−16−851q−18 + 637q−20 + 1410q−22 + 603q−24−1234q−26−1355q−28 + 44q−30 + 1375q−32 + 1184q−34−661q−36−1371q−38−596q−40 + 798q−42 + 1277q−44 + 35q−46−841q−48−795q−50 + 113q−52 + 829q−54 + 344q−56−223q−58−520q−60−192q−62 + 306q−64 + 235q−66 + 53q−68−179q−70−143q−72 + 57q−74 + 65q−76 + 55q−78−33q−80−46q−82 + 10q−84 + 6q−86 + 16q−88−5q−90−10q−92 + 4q−94 + 3q−98−q−100−2q−102 + q−104
5 −q155 + 2q153 + q151−3q149−q143 + 5q141 + 4q139−12q137−9q135 + 7q133 + 17q131 + 24q129 + 2q127−49q125−75q123−12q121 + 101q119 + 158q117 + 78q115−145q113−337q111−254q109 + 171q107 + 604q105 + 589q103−39q101−912q99−1202q97−373q95 + 1165q93 + 2062q91 + 1214q89−1099q87−3064q85−2613q83 + 473q81 + 3942q79 + 4493q77 + 902q75−4300q73−6533q71−3114q69 + 3789q67 + 8352q65 + 5868q63−2305q61−9333q59−8700q57−148q55 + 9257q53 + 11041q51 + 3042q49−8010q47−12339q45−5899q43 + 5830q41 + 12446q39 + 8135q37−3201q35−11407q33−9408q31 + 579q29 + 9517q27 + 9738q25 + 1643q23−7232q21−9264q19−3313q17 + 4893q15 + 8320q13 + 4522q11−2747q9−7229q7−5426q5 + 827q3 + 6187q + 6247q−1 + 985q−3−5254q−5−7142q−7−2811q−9 + 4318q−11 + 8093q−13 + 4802q−15−3177q−17−8965q−19−6968q−21 + 1664q−23 + 9472q−25 + 9135q−27 + 357q−29−9294q−31−11025q−33−2813q−35 + 8255q−37 + 12209q−39 + 5389q−41−6250q−43−12367q−45−7686q−47 + 3530q−49 + 11371q−51 + 9196q−53−591q−55−9250q−57−9606q−59−2079q−61 + 6469q−63 + 8899q−65 + 3873q−67−3574q−69−7197q−71−4670q−73 + 1100q−75 + 5092q−77 + 4462q−79 + 541q−81−3006q−83−3577q−85−1341q−87 + 1376q−89 + 2443q−91 + 1448q−93−362q−95−1411q−97−1141q−99−124q−101 + 664q−103 + 741q−105 + 256q−107−257q−109−396q−111−201q−113 + 61q−115 + 174q−117 + 125q−119 + q−121−75q−123−56q−125−3q−127 + 23q−129 + 18q−131 + 8q−133−9q−135−12q−137 + 4q−139 + 5q−141−q−143−3q−149 + q−151 + 2q−153−q−155

A2 Invariants.

Weight Invariant
1,0 −q16 + q12−2q10 + 3q8 + q6−q4 + 2q2−3 + 2q−2−q−4 + q−6 + 3q−8−2q−10 + q−12−q−16
2,0 q42−2q38−q36 + 3q34 + 2q32−6q30−3q28 + 10q26 + 2q24−13q22−3q20 + 15q18 + 5q16−17q14 + 2q12 + 14q10−6q8−10q6 + 6q4 + 4q2−6 + 6q−2 + 9q−4−6q−6−4q−8 + 14q−10−19q−14 + 3q−16 + 14q−18−4q−20−12q−22 + 4q−24 + 11q−26−3q−28−7q−30 + 2q−32 + 3q−34−q−36−2q−38 + q−42

A3 Invariants.

Weight Invariant
0,1,0 q34−2q32 + q30 + 4q28−9q26 + 2q24 + 11q22−19q20 + 3q18 + 21q16−23q14 + 3q12 + 23q10−16q8−4q6 + 13q4−2q2−8−2q−2 + 13q−4−4q−6−16q−8 + 23q−10 + 3q−12−23q−14 + 21q−16 + 3q−18−19q−20 + 11q−22 + 2q−24−9q−26 + 4q−28 + q−30−2q−32 + q−34
1,0,0 −q21−q17 + q15−2q13 + 4q11 + 3q7−q5 + q3−2q−2q−1 + q−3−q−5 + 3q−7 + 4q−11−2q−13 + q−15−q−17−q−21

B2 Invariants.

Weight Invariant
0,1 −q34 + 2q32−5q30 + 8q28−13q26 + 18q24−23q22 + 27q20−27q18 + 25q16−17q14 + 9q12 + 5q10−18q8 + 32q6−43q4 + 50q2−54 + 50q−2−43q−4 + 32q−6−18q−8 + 5q−10 + 9q−12−17q−14 + 25q−16−27q−18 + 27q−20−23q−22 + 18q−24−13q−26 + 8q−28−5q−30 + 2q−32−q−34
1,0 q56−2q52−2q50 + 3q48 + 6q46−q44−11q42−7q40 + 11q38 + 17q36−4q34−25q32−10q30 + 23q28 + 24q26−12q24−29q22−2q20 + 28q18 + 13q16−19q14−17q12 + 12q10 + 18q8−6q6−18q4 + 2q2 + 19 + 2q−2−18q−4−6q−6 + 18q−8 + 12q−10−17q−12−19q−14 + 13q−16 + 28q−18−2q−20−29q−22−12q−24 + 24q−26 + 23q−28−10q−30−25q−32−4q−34 + 17q−36 + 11q−38−7q−40−11q−42−q−44 + 6q−46 + 3q−48−2q−50−2q−52 + q−56

G2 Invariants.

Weight Invariant
1,0 q80−2q78 + 5q76−8q74 + 8q72−7q70−2q68 + 16q66−32q64 + 47q62−52q60 + 36q58−3q56−48q54 + 102q52−137q50 + 132q48−84q46−6q44 + 105q42−179q40 + 206q38−155q36 + 56q34 + 61q32−147q30 + 164q28−107q26 + 10q24 + 90q22−135q20 + 110q18−14q16−110q14 + 208q12−235q10 + 166q8−34q6−128q4 + 254q2−299 + 253q−2−128q−4−32q−6 + 166q−8−233q−10 + 206q−12−108q−14−12q−16 + 109q−18−135q−20 + 90q−22 + 10q−24−107q−26 + 164q−28−150q−30 + 63q−32 + 55q−34−156q−36 + 206q−38−180q−40 + 106q−42−6q−44−84q−46 + 132q−48−136q−50 + 102q−52−47q−54−4q−56 + 36q−58−51q−60 + 46q−62−32q−64 + 16q−66−2q−68−7q−70 + 8q−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 71"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t3 + 7t2−18t + 25−18t−1 + 7t−2−t−3
In[5]:= Conway[K][z]
Out[5]= −z6 + z4 + z2 + 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]= { 77, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q5 + 3q4−6q3 + 10q2−12q + 13−12q−1 + 10q−2−6q−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−2−3z4−a4z2 + 4a2z2 + 4z2a−2−z2a−4−5z2−a4 + 3a2 + 3a−2−a−4−3
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= az9 + z9a−1 + 3a2z8 + 3z8a−2 + 6z8 + 4a3z7 + 8az7 + 8z7a−1 + 4z7a−3 + 3a4z6 + 2a2z6 + 2z6a−2 + 3z6a−4−2z6 + a5z5−5a3z5−15az5−15z5a−1−5z5a−3 + z5a−5−6a4z4−12a2z4−12z4a−2−6z4a−4−12z4−2a5z3 + 7az3 + 7z3a−1−2z3a−5 + 4a4z2 + 10a2z2 + 10z2a−2 + 4z2a−4 + 12z2 + a5z + a3z−az−za−1 + za−3 + za−5−a4−3a2−3a−2−a−4−3

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

Same Alexander/Conway Polynomial: {K11n156, K11n179,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_104,}

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 71"];
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 + 7t2−18t + 25−18t−1 + 7t−2−t−3, −q5 + 3q4−6q3 + 10q2−12q + 13−12q−1 + 10q−2−6q−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]= {K11n156, K11n179,}
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_104,}

[edit] Vassiliev invariants

V2 and V3: (1, 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
4 0 8 \frac{62}{3} -\frac{14}{3} 0 0 0 0 \frac{32}{3} 0 \frac{248}{3} -\frac{56}{3} \frac{3631}{30} \frac{326}{5} -\frac{2578}{45} \frac{209}{18} -\frac{1169}{30}

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 71. 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        41 -3
5       62  4
3      64   -2
1     76    1
-1    67     1
-3   46      -2
-5  26       4
-7 14        -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r = 1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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 + q13 + 9q12−17q11 + q10 + 37q9−47q8−12q7 + 89q6−77q5−42q4 + 140q3−87q2−73q + 161−73q−1−87q−2 + 140q−3−42q−4−77q−5 + 89q−6−12q−7−47q−8 + 37q−9 + q−10−17q−11 + 9q−12 + q−13−3q−14 + q−15
3 −q30 + 3q29−q28−4q27−2q26 + 14q25 + 2q24−29q23−8q22 + 57q21 + 24q20−98q19−62q18 + 153q17 + 126q16−209q15−220q14 + 249q13 + 352q12−282q11−485q10 + 271q9 + 633q8−245q7−754q6 + 186q5 + 859q4−124q3−914q2 + 41q + 941 + 33q−1−914q−2−116q−3 + 859q−4 + 178q−5−753q−6−238q−7 + 631q−8 + 264q−9−482q−10−275q−11 + 349q−12 + 243q−13−218q−14−203q−15 + 124q−16 + 149q−17−62q−18−96q−19 + 25q−20 + 56q−21−8q−22−29q−23 + 2q−24 + 14q−25−2q−26−4q−27−q−28 + 3q−29−q−30
4 q50−3q49 + q48 + 4q47−3q46 + 5q45−17q44 + 6q43 + 25q42−13q41 + 9q40−73q39 + 19q38 + 113q37−3q36 + q35−273q34−17q33 + 345q32 + 179q31 + 72q30−771q29−345q28 + 642q27 + 746q26 + 557q25−1487q24−1253q23 + 596q22 + 1622q21 + 1799q20−1966q19−2647q18−179q17 + 2332q16 + 3644q15−1775q14−3978q13−1578q12 + 2453q11 + 5481q10−968q9−4751q8−3066q7 + 1993q6 + 6730q5 + 77q4−4819q3−4190q2 + 1181q + 7163 + 1074q−1−4252q−2−4763q−3 + 180q−4 + 6734q−5 + 1896q−6−3115q−7−4696q−8−885q−9 + 5468q−10 + 2357q−11−1602q−12−3902q−13−1700q−14 + 3609q−15 + 2218q−16−201q−17−2556q−18−1876q−19 + 1777q−20 + 1507q−21 + 549q−22−1192q−23−1399q−24 + 572q−25 + 676q−26 + 590q−27−335q−28−724q−29 + 99q−30 + 162q−31 + 321q−32−26q−33−264q−34 + 12q−35−2q−36 + 110q−37 + 16q−38−73q−39 + 10q−40−13q−41 + 25q−42 + 6q−43−17q−44 + 5q−45−3q−46 + 4q−47 + q−48−3q−49 + q−50
5 −q75 + 3q74−q73−4q72 + 3q71−2q69 + 9q68−2q67−20q66 + 6q65 + 17q64 + 8q63 + 14q62−28q61−73q60−13q59 + 93q58 + 132q57 + 63q56−141q55−335q54−208q53 + 232q52 + 645q51 + 548q50−218q49−1123q48−1225q47−38q46 + 1694q45 + 2358q44 + 777q43−2190q42−3942q41−2274q40 + 2265q39 + 5905q38 + 4698q37−1560q36−7934q35−8044q34−262q33 + 9528q32 + 12145q31 + 3466q30−10364q29−16592q28−7789q27 + 9884q26 + 20804q25 + 13253q24−8189q23−24433q22−19005q21 + 5203q20 + 26921q19 + 24892q18−1369q17−28387q16−30073q15−3009q14 + 28652q13 + 34543q12 + 7409q11−28050q10−37881q9−11641q8 + 26655q7 + 40331q6 + 15388q5−24759q4−41656q3−18770q2 + 22324q + 42219 + 21627q−1−19497q−2−41716q−3−24130q−4 + 16071q−5 + 40416q−6 + 26109q−7−12228q−8−37918q−9−27557q−10 + 7865q−11 + 34465q−12 + 28141q−13−3353q−14−29823q−15−27778q−16−1073q−17 + 24478q−18 + 26142q−19 + 4853q−20−18487q−21−23467q−22−7689q−23 + 12749q−24 + 19702q−25 + 9182q−26−7435q−27−15468q−28−9473q−29 + 3344q−30 + 11150q−31 + 8549q−32−407q−33−7295q−34−6989q−35−1219q−36 + 4247q−37 + 5130q−38 + 1826q−39−2093q−40−3398q−41−1770q−42 + 778q−43 + 2044q−44 + 1375q−45−128q−46−1085q−47−922q−48−119q−49 + 506q−50 + 546q−51 + 162q−52−212q−53−294q−54−104q−55 + 73q−56 + 121q−57 + 79q−58−20q−59−71q−60−24q−61 + 16q−62 + 8q−63 + 16q−64 + 6q−65−20q−66−2q−67 + 9q−68−2q−69 + 3q−71−4q−72−q−73 + 3q−74−q−75
6 q105−3q104 + q103 + 4q102−3q101−3q99 + 10q98−13q97−3q96 + 27q95−16q94−11q93−18q92 + 44q91−17q90−4q89 + 93q88−59q87−94q86−120q85 + 122q84 + 43q83 + 129q82 + 374q81−127q80−445q79−704q78−23q77 + 163q76 + 820q75 + 1679q74 + 417q73−1100q72−2754q71−1860q70−890q69 + 2140q68 + 5766q67 + 4320q66 + 105q65−6429q64−8228q63−7743q62 + 486q61 + 12671q60 + 16071q59 + 10267q58−6146q57−18829q56−26781q55−14288q54 + 14126q53 + 34903q52 + 37649q51 + 11689q50−22346q49−56318q48−52069q47−6912q46 + 46334q45 + 79069q44 + 58082q43 + 1062q42−78635q41−107682q40−61642q39 + 28514q38 + 113841q37 + 125521q36 + 61506q35−70706q34−157830q33−139732q32−27071q31 + 118633q30 + 188775q29 + 145658q28−25843q27−179362q26−214439q25−105180q24 + 88395q23 + 225359q22 + 226146q21 + 40067q20−168064q19−264145q18−179823q17 + 38263q16 + 231417q15 + 283013q14 + 103990q13−136883q12−285072q11−234321q10−12878q9 + 217193q8 + 313167q7 + 153420q6−99859q5−284929q4−267400q3−56989q2 + 192187q + 322667 + 189245q−1−60743q−2−269454q−3−284321q−4−96924q−5 + 156541q−6 + 314357q−7 + 215957q−8−15153q−9−235908q−10−285368q−11−135952q−12 + 104928q−13 + 283046q−14 + 230811q−15 + 38316q−16−178507q−17−262541q−18−167495q−19 + 38345q−20 + 222561q−21 + 222168q−22 + 88504q−23−100992q−24−208570q−25−175561q−26−27261q−27 + 139141q−28 + 180916q−29 + 114899q−30−23730q−31−131093q−32−149140q−33−67729q−34 + 56486q−35 + 115371q−36 + 105182q−37 + 26906q−38−55236q−39−97296q−40−70861q−41 + 1388q−42 + 50932q−43 + 69344q−44 + 40128q−45−6076q−46−45190q−47−48017q−48−17786q−49 + 10095q−50 + 31626q−51 + 28508q−52 + 11165q−53−12553q−54−22005q−55−14284q−56−4461q−57 + 8682q−58 + 12688q−59 + 9763q−60−403q−61−6499q−62−6038q−63−4750q−64 + 488q−65 + 3493q−66 + 4503q−67 + 1237q−68−1008q−69−1368q−70−2102q−71−677q−72 + 440q−73 + 1427q−74 + 509q−75−6q−76−17q−77−589q−78−320q−79−67q−80 + 366q−81 + 81q−82 + 3q−83 + 104q−84−114q−85−79q−86−49q−87 + 97q−88−7q−89−20q−90 + 42q−91−18q−92−10q−93−16q−94 + 27q−95−3q−96−13q−97 + 10q−98−3q−99−3q−101 + 4q−102 + q−103−3q−104 + q−105
7 −q140 + 3q139−q138−4q137 + 3q136 + 3q134−5q133−6q132 + 18q131−4q130−17q129 + 10q128 + 5q127 + 19q126−22q125−49q124 + 46q123−q122−25q121 + 50q120 + 35q119 + 95q118−67q117−237q116−40q115−72q114 + 13q113 + 305q112 + 327q111 + 499q110−6q109−834q108−835q107−981q106−329q105 + 1035q104 + 1848q103 + 2739q102 + 1559q101−1414q100−3589q99−5583q98−4419q97 + 249q96 + 5324q95 + 10843q94 + 10858q93 + 3756q92−6224q91−18002q90−22048q89−13892q88 + 2682q87 + 25611q86 + 39508q85 + 33822q84 + 10089q83−29584q82−61860q81−66529q80−38785q79 + 22063q78 + 84042q77 + 112851q76 + 90294q75 + 7196q74−96800q73−168206q72−167942q71−69313q70 + 85394q69 + 221631q68 + 270081q67 + 172974q66−34028q65−256579q64−386292q63−319674q62−71005q61 + 252161q60 + 498162q59 + 502190q58 + 237245q57−189446q56−583385q55−703245q54−460322q53 + 56550q52 + 617416q51 + 897896q50 + 726787q49 + 149273q48−583820q47−1061619q46−1012012q45−415748q44 + 474654q43 + 1169976q42 + 1289213q41 + 724519q40−294998q39−1212370q38−1533060q37−1046266q36 + 60784q35 + 1184229q34 + 1724267q33 + 1357122q32 + 206118q31−1096577q30−1854829q29−1633207q28−480434q27 + 963299q26 + 1925041q25 + 1862512q24 + 741680q23−804916q22−1944029q21−2038995q20−974677q19 + 638143q18 + 1924202q17 + 2166570q16 + 1172133q15−477017q14−1878611q13−2252020q12−1334234q11 + 327516q10 + 1818177q9 + 2306060q8 + 1465819q7−191775q6−1748494q5−2335919q4−1575157q3 + 63986q2 + 1670691q + 2348802 + 1669944q−1 + 61475q−2−1580758q−3−2343857q−4−1755851q−5−194546q−6 + 1471175q−7 + 2319234q−8 + 1834551q−9 + 339718q−10−1333529q−11−2265344q−12−1901929q−13−500698q−14 + 1159908q−15 + 2173423q−16 + 1949453q−17 + 672252q−18−947289q−19−2032365q−20−1963836q−21−844916q−22 + 698683q−23 + 1836854q−24 + 1930889q−25 + 1001339q−26−425443q−27−1586172q−28−1839147q−29−1122928q−30 + 146973q−31 + 1291084q−32 + 1682678q−33 + 1189349q−34 + 113341q−35−968596q−36−1466308q−37−1189743q−38−329709q−39 + 646595q−40 + 1203823q−41 + 1118353q−42 + 482769q−43−350263q−44−919307q−45−986055q−46−561821q−47 + 106912q−48 + 640217q−49 + 808886q−50 + 568347q−51 + 70019q−52−391940q−53−613950q−54−516053q−55−174862q−56 + 194438q−57 + 425070q−58 + 424638q−59 + 216201q−60−55025q−61−263003q−62−318303q−63−209546q−64−27811q−65 + 139438q−66 + 216128q−67 + 173949q−68 + 65113q−69−56221q−70−131497q−71−127792q−72−71267q−73 + 8490q−74 + 70271q−75 + 83625q−76 + 60143q−77 + 13260q−78−31096q−79−48559q−80−43446q−81−18945q−82 + 9772q−83 + 24774q−84 + 27496q−85 + 16498q−86−164q−87−10549q−88−15397q−89−11692q−90−2922q−91 + 3329q−92 + 7818q−93 + 7170q−94 + 2757q−95−364q−96−3350q−97−3768q−98−2015q−99−706q−100 + 1365q−101 + 1990q−102 + 1035q−103 + 541q−104−395q−105−735q−106−483q−107−569q−108 + 88q−109 + 442q−110 + 196q−111 + 192q−112−44q−113−70q−114−q−115−195q−116−42q−117 + 98q−118 + 25q−119 + 39q−120−32q−121−4q−122 + 48q−123−46q−124−20q−125 + 19q−126 + 4q−127 + 10q−128−17q−129−4q−130 + 18q−131−6q−132−5q−133 + 3q−134 + 3q−136−4q−137−q−138 + 3q−139−q−140

[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_70

10_72

Retrieved from "http://katlas.math.toronto.edu/wiki/10_71"
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